11institutetext: Shanghai Astronomical Observatory, Key Laboratory of Radio
Astronomy, CAS, 80 Nandan Road, Shanghai 200030, China
22institutetext: School of Astronomy and Space Sciences, University of Chinese
Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China
33institutetext: Key Laboratory of Radio Astronomy and Technology, Chinese
Academy of Sciences, A20 Datun Road, Beijing, 100101, P. R. China
44institutetext: Space Research Institute of RAS,Profsoyusnaya street, 84/32,
117997, Russia
55institutetext: Special Astrophysical Observatory of RAS, Nizhny Arkhyz,
369167, Russia
66institutetext: Kazan Federal University, 18 Kremlyovskaya St, Kazan 420008,
Russia
77institutetext: Institute of Applied Astronomy RAS, St. Petersburg 191187,
Russia

# VLBI observations of the high-redshift X-ray bright blazar SRGE
J170245.3+130104

Yuanqi Liu 11 Tao An<EMAIL_ADDRESS>Shaoguang Guo 112233 Yingkang
Zhang 1133 Ailing Wang 1122 Zhijun Xu 11 Georgii Khorunzhev 44 Yulia Sotnikova
55 Timur Mufakharov 5566 Alexander Mikhailov 55 Marat Mingaliev 556677

(Received 30/01/2024; accepted 26/02/2024)

###### Abstract

Aims. The X-ray luminous and radio-loud AGN SRGE J170245.3+130104 discovered
at $z\sim 5.5$ provides unique chances to probe the SMBH growth and evolution
with powerful jets in the early Universe.

Methods. We present $1.35-5.1$ GHz Very Long Baseline Array (VLBA) results on
the radio continuum emission and spectrum analysis for this quasar in a low
flux density state.

Results. This source is unresolved at three frequencies with the total flux
densities of $8.35\pm 0.09\ \rm mJy\ beam^{-1}$, $7.47\pm 0.08\ \rm mJy\
beam^{-1}$, and $6.57\pm 0.02\ \rm mJy\ beam^{-1}$ at 1.73 GHz, 2.26 GHz, and
4.87 GHz, respectively. Meanwhile, the brightness temperature is higher than
$10^{9}$ K.

Conclusions. Compared with previous radio observations with arcsec-scale
resolution, nearly all the radio emission from this source concentrates in the
very central milli-arcsecond (mas) scale area. We confirm this source is a
bright blazar at $z>5$. This young AGN provide us the great chances to
understand the first generation of strong jets in the early Universe.

###### Key Words.:

radio continuum emission – early Universe – Quasars

##  1 Introduction

The study of high-redshift active galactic nuclei (AGNs) plays a crucial role
in advancing our understanding of the early Universe, particularly in
exploring the formation and evolution of supermassive black holes (SMBHs).
These distant AGNs provide a unique window into the nascent stages of SMBHs,
shedding light on their accretion processes and immediate environment. Such
studies are crucial for understanding the mechanisms that drive the early
growth of these cosmic monsters.

Currently, around 600 AGNs have been discovered at redshift higher than 5 (for
example, Fan et al. 2006; Willott et al. 2010; Yang et al. 2017, 2023), which
is close to the end of cosmic reionization. These discoveries are essential
for understanding this crucial moment in the evolution of the Universe.

Blazars are a subclass of AGN with jets aligned close to the line of sight,
whose emission is relativistic beamed and Doppler boosted, providing favorable
conditions for observing high-redshift AGN. Observations of jet emission from
high-$z$ blazars enable tests of relativistic jet models and physical
properties (such as jet viewing angle and Lorentz factor) under extreme
conditions in the early Universe and constraints on jet power and particle
acceleration. Statistical studies of the number density, distribution, and
evolution of blazars can yield insights into the co-evolution of black holes
and galaxies in the early Universe. To date, PSO J030947.49+271757.31 at
$z\sim 6.1$ (hereafter J0309+2717) is the highest redshift blazar discovered
(Belladitta et al., 2020). Since very few blazars have been observed at $z>5$,
discovering and studying more high-$z$ ($z>5$) blazars is extremely valuable
for enriching our understanding of high-redshift AGN.

Our research focuses on SRGE J170245.3+130104 (hereafter J1702+1301), one of
the highest-redshift and high-luminosity X-ray quasars recently discovered by
the eROSITA telescope on-board the Spectrum Roentgen Gamma (SRG; Merloni et
al. 2020) during its first half-year X-ray All-sky Survey (Khorunzhev et al.,
2021). Follow-up the Burst Alert Telescope (BAT) telescope’s spectroscopic
observations measured the redshift of J1702+1301 to be $z=5.466\pm 0.003$, and
although only a limited spectral coverage has been obtained so far, the mass
of its central black hole is estimated to be at least $10^{8}M_{\odot}$
(Khorunzhev et al., 2021). J1702+1301 exhibits powerful X-ray and radio
emission with a radio luminosity of $10^{27}\leavevmode\nobreak\ \rm
W\leavevmode\nobreak\ Hz^{-1}$, a radio loudness of $R>1100$, a rapidly
varying radio flux density, and a flat radio spectrum (An et al., 2023). All
these observations are consistent with those of blazars, suggesting that
J1702+1301 is most likely a high-$z$ blazar with a relativistic jet aligned to
the line of sight. J1702+1301 presents an exceptional opportunity to
investigate the first generation of strong jets emanating from young AGN.

J1702+1301 stands out due to its extraordinary luminosity in X-ray and radio
at $z>5$, making it a benchmark for follow-up observational campaigns. Its
extreme characteristics make it an invaluable reference point for comparative
studies of similar objects, enhancing our overall knowledge of high-redshift
AGNs. Our Very Long Baseline Array (VLBA) observations at 1.73 GHz, 2.26 GHz,
and 4.87 GHz have revealed that nearly all radio emissions from J1702+1301 are
concentrated in a central milli-arcsecond region, with a brightness
temperature exceeding $10^{9}$ K (An et al., 2023). This concentration of
emissions in a compact region suggests a highly active central core, a typical
characteristic of blazars. Moreover, our observations of J1702+1301 at these
frequencies, corresponding to significantly higher rest-frame frequencies due
to the source’s redshift, enable a deeper understanding of the blazar’s
properties and behavior in the early universe. By comparing J1702+1301’s
observational data with that of other high-redshift blazars, we aim to
construct a more comprehensive picture of blazar evolution and jet formation
in the nascent universe. Understanding the accretion dynamics in these early
stages is essential for a comprehensive view of SMBH growth and energy output.
Through our detailed analysis of J1702+1301, we hope to provide valuable
insights into the development of strong jets and the role of blazars in cosmic
evolution, thus advancing the field of high-redshift AGN research and
contributing to the broader picture of cosmic evolution.

Throughout this work, we adopt a $\rm\Lambda CDM$ cosmology with $\rm
H_{0}=70\ km\ s^{-1}Mpc^{-1}$, $\Omega_{\rm m}=0.3$ and
$\Omega_{\Lambda}=0.7$. With this cosmological model, 1$\arcsec$ at the
redshift of J1702+1301 corresponds to 5.98 kpc.

##  2 observations

The VLBA observations of J1702+1301 were conducted in December 2021 at L, S,
and C bands. The VLBA observations of J1702+1301 were conducted in December
2021 at L, S, and C bands that cover the observing frequencies from 1.35 to
7.9 GHz (An et al., 2023). At the redshift of J1702+1301, the central
frequencies of these bands correspond to rest-frame frequencies of 11.3 GHz,
14.7 GHz, and 31.7 GHz, respectively, allowing us to probe the blazar’s
properties at higher GHz frequencies.

A notable challenge during these observations was the relatively low flux
density state of this blazar (An et al., 2023), which impacted our ability to
detect extended jet emission. This state of J1702+1301, compared to its other
observing epochs, provided a unique opportunity to study the blazar’s
characteristics in its quiescent phase. Such observations are crucial for
understanding the variability and typical emission properties of high-redshift
blazars.

The data were recorded with dual polarizations and sampled at two bits, using
4 spectral windows in each band, further split into 128 channels at L and S
bands or 256 baseband channels at C band with a channel width of 500 kHz. The
observations were correlated at the VLBA correlator of the US National Radio
Astronomy Observatory in Socorro, NM, with a correlator integration time of
2s. Table 1 summarizes the observing parameters.

For calibration purposes, we selected J1706+1208 and J1642+3948 as fringe
finders and J1707+1331 (1.3 degree away from the target, $S_{\rm S-band}\sim$
0.3 Jy) as the phase calibrator for observations at the three frequencies. Due
to the target’s limited brightness, we employed phase-referencing mode with a
cycle comprising 1-minute on the phase calibrator and 3.5-minute observing
time on the target source. Fringe finders were observed once an hour. During
the data reduction, we used J1706+1208 for fringe fitting, compensating for
the unavailability of two antennas in the observations of J1642+3948.

Data reduction and analysis were performed with the Astronomical Image
Processing System (AIPS; Greisen 2003) of the NRAO. We use the ParselTongue
pipeline 111https://github.com/SHAO-SKA/vlbi-pipeline deployed at the China
SKA Regional Centre (An et al., 2019, 2022) to automatically achieve the
standard calibration steps, mainly including instrumental delay, ionospheric
corrections, and fringe fitting. The pipeline also applied automatic flagging
to avoid the effect of Radio Frequency Interference (RFI) and other bad data.
We export the calibrated data of calibrators to the Caltech Difmap (Shepherd,
1997) package for imaging and CLEAN task of the calibrators. Then the self-
calibration for both phase and amplitude was performed on the phase calibrator
based on its clean model. These self-calibration solutions were applied to the
target. The overall calibration uncertainties are less than 5%. As the final
step, we applied mapping and clean procedure on the target J1702+1301 in the
natural weighting. The intensity maps are shown in Fig. 1.

Table 1: Observation information and image parameters. Parameters | L band | S band | C band
---|---|---|---
Observing date (in 2021) | Dec 06 | Dec 07 | Dec 21
Total observing time (hr) | 2.75 | 4 | 2.75
Central Frequency (GHz) | 1.73 | 2.26 | 4.87
Total bandwidth (MHz) | 256 | 256 | 512
Synthesis beam (mas) | $11.73\times 5.19$ | $9.46\times 5.10$ | $10.91\times 4.46$
Image rms noise level (mJy beam-1) | 0.11 | 0.12 | 0.42
Peak flux density (mJy beam-1) | 6.77 | 6.85 | 6.40
Integrated flux density (mJy) | 8.35 $\pm$ 0.09 | 7.47 $\pm$ 0.08 | 6.57 $\pm$ 0.02
Size (mas) | $3.276\pm 0.043$ | $1.626\pm 0.068$ | $0.905\pm 0.006$
Brightness temperature ($10^{9}$K) | 2.0 | 4.4 | 2.7
Note: natural weighting is used in all the images. |  |  |

Figure 1: The VLBA radio continuum images of J1702+1301. The left, middle, and
right panel shows the observations at 1.73 GHz, 2.36 GHz, and 4.87 GHz,
respectively. The images are all centered at RA: 17h02m45.3119s, Dec:
+13d01m02.222s, which is 22.3 mas away southeast to the measured center from
optical/X-ray observations shown as a white cross. The contour levels are
[-1,1,2,4,8,16,32,64]*3$\sigma$, where the $\sigma$ corresponds to the rms
noise level from each image. The synthesis beam for each image is shown at the
lower-left corner.

##  3 Results

We successfully detected J1702+1301 at all three observing frequencies (1.73
GHz, 2.26 GHz, and 4.87 GHz) with remarkable signal-to-noise ratios ranging
from 15 to 62. These observations were critical in determining the compact
nature of the emission sources. The total intensity maps show a single,
unresolved component at all frequencies (Fig. 1). This observational result is
consistent across a broad field of view spanning 1–2 arcsec, with no
additional features detected above a $5\sigma$ threshold ($\sigma$ being the
image rms noise). The precise equatorial coordinates of the emission peak are
RA: 17h02m45.3119s, dec: +13d01m02.222s, with astrometric errors of $\sim$0.1
mas arising mainly from uncertainties in the phase reference calibrator’s
position. The VLBI position derived at the three frequencies is consistent
within the error margin. The previously reported optical/X-ray coordinates are
RA: 17h02m45.3s, Dec: +13d01m02.24s, which are 22.3 mas northwest of the VLBI
position. This compact VLBI component represents the jet base close to the
central SMBH (see more discussion below). The radio–optical positional offset
can plausibly be explained by astrometric uncertainties in the optical/X-ray
measurements due to the weakness of the source in optical and X-ray images.

We fit circular Gaussian models to the J1702+1301 VLBI source using Difmap’s
modelfit program. The measured total flux densities of the source were 8.35
$\pm$ 0.09 mJy, 7.47 $\pm$ 0.08 mJy, and 6.57 $\pm$ 0.02 mJy at observation
frequencies of 1.73 GHz, 2.26 GHz, and 4.87 GHz, respectively (see Table 1).
The fitted Gaussian source sizes (FWHM) are $3.276\pm 0.043$ mas, $1.626\pm
0.068$ mas, and $0.905\pm 0.006$ mas at the three frequencies, respectively.
At 4.87 GHz the fitted peak flux density is consistent with the integrated
flux density, indicating that the emission originates from a component too
compact to be resolved by the VLBA. However, at the lower frequencies of 1.73
and 2.26 GHz, our detection is limited to a single unresolved core, with
integrated flux densities slightly exceeding peak flux densities. The current
data set, while extensive, does not allow for a conclusive determination of
whether jet emission is mixed with the core emission at 1.73 and 2.26 GHz. To
unravel the complexities of J1702+1301’s jet morphology and its potential
evolution, further investigations with higher-resolution observations are
essential.

With the measured flux density and source size, we calculated the brightness
temperature, which is a crucial parameter in understanding the physical
conditions of the emitting region. Employing the established formula for
brightness temperature calculation from Condon et al. (1982):

$T_{b}=1.22\times 10^{12}\left(1+z\right)\frac{S_{\nu}}{\nu^{2}\theta^{2}},$
(1)

where $\nu$ is the observing frequency in GHz, $S_{\nu}$ denotes the surface
brightness in $\rm Jy\ beam^{-1}$, and $\theta$ represents the source size at
different frequency. We determined that the brightness temperature of
J1702+1301 is $\sim 10^{9}$K. This measurement is particularly significant as
it provides compelling evidence of the non-thermal nature of the emission. It
is important to note that since the source is only marginally resolved in our
observations, the fitted source sizes can be regarded as upper limits, and
thus the derived brightness temperatures should be treated as lower limits.

The high brightness temperatures, exceeding $10^{9}$ K at all three
frequencies, provide robust evidence that the emission has a non-thermal
origin. This is indicative of processes such as synchrotron emission, which
are common in AGN and especially pronounced in blazars. The temperature values
observed for J1702+1301, although not as extreme as some blazars in the low-
redshift universe, are still remarkable for a high-redshift object and
consistent well with the expected properties of blazars at such distances (for
example, Cao et al. 2014; Frey et al. 2011; Zhang et al. 2022; Spingola et al.
2020). Note that these measurements are obtained at rest-frame frequencies of
10–30 GHz, where a general trend of decreasing brightness temperature with
increasing frequency has been observed (Cheng et al., 2020). These
temperatures reinforce the interpretation of J1702+1301 as a blazar
(Khorunzhev et al., 2021; An et al., 2023), characterized by a highly
energetic and active nucleus.

The spectral energy distribution (SED) analysis of J1702+1301 is crucial for
understanding the radiative mechanisms and physical conditions within the
blazar. To construct the SED, we included data from our VLBA observations
(represented by red squares in Fig. 2) alongside archival data from the Giant
Metrewave Radio Telescope (GMRT), the Australian Square Kilometre Array
Pathfinder (ASKAP), and the Very Large Array Sky Survey (VLASS) (indicated by
blue labels in Fig. 2). Our analysis employed a standard non-thermal power-law
spectrum to model the radio continuum, fitting the data across the three VLBA
frequencies (corresponding to rest-frame frequencies of 9.75–33.15 GHz at the
source’s redshift). This approach is rooted in the form
$S_{\nu}\sim\nu^{\alpha}$, where $\alpha$ is the spectral index, and $S_{\nu}$
is the flux density at frequency $\nu$. Through a least-squares fitting
method, we determined the spectral index to be $\alpha=-0.21\pm 0.001$. The
resulting spectrum, illustrated by the red line in Fig. 2, reveals a slightly
steeper spectral index than previous results; these earlier results,
represented by the black dashed line, were derived from synthesis imaging at
much lower resolutions over 0.3–3 GHz (An et al., 2023). The fitted spectral
index of $\alpha=-0.21$ is indicative of a flat-spectrum radio core, a
characteristic commonly associated with blazars. This finding agrees well with
the high brightness temperatures observed and the compact nature of the VLBI-
detected radio core, providing compelling evidence of J1702+1301 being a
blazar with a relativistic jet oriented close to our line of sight.

##  4 discussion

The VLBA observations of J1702+1301 provide valuable insights into the nature
of this high-redshift AGN. The detected compact VLBI component, in close
proximity to the central SMBH, likely represents the base of the jet.

The total flux densities measured at the observing frequencies (1.73 GHz, 2.26
GHz, and 4.87 GHz) were 8.35 $\pm$ 0.09 mJy, 7.47 $\pm$ 0.08 mJy, and 6.57
$\pm$ 0.02 mJy, respectively. The consistent flux densities across the
observed frequencies are indicative of a compact, high-brightness temperature
source, a characteristic feature of blazars. The spectral index,
$\alpha=-0.21$, is consistent with expectations for a flat-spectrum radio
core, reinforcing the blazar identification of J1702+1301.

One notable aspect of our study is the positional offset between the radio and
optical/X-ray coordinates of J1702+1301. The positional offset of 22.3 mas
between the radio and optical/X-ray coordinates is significant (see Section
3). While astrometric uncertainties in the optical and X-ray bands could
partly account for this discrepancy, particularly given the source’s relative
weakness in these bands, this substantial offset is uncommon in blazar
observations. Radio-optical offsets in other blazars are usually much smaller.
This discrepancy may indicate more complex jet dynamics than typically
observed or could suggest variations in the regions of emission.

The nature of the compact VLBI component, especially its high brightness
temperature and flat radio spectrum, reinforce the significance of J1702+1301
as a blazar. This conclusion is consistent with the observed characteristics
of other high-redshift blazars and contributes to our broader understanding of
AGN jet formation and evolution in the early universe.

The RATAN-600 telescope has been monitoring J1702+1301 since late 2020 at 4.7,
8.2, and 11.2 GHz as shown in Fig.5 in An et al. 2023. According to the flux
density curve, we note that our VLBA observations were carried out when
J1702+1301 was in a relatively low flux density state compared to the active
state after February 2022. This timing likely explains the absence of extended
jet emission in our VLBI images. Such a state provides a baseline for the
blazar’s ‘normal’ condition. Given J1702+1301’s high brightness temperature
and rapid variability, upcoming VLBI observations during its flaring state are
highly anticipated. Comparing these future observations with our current data
will enhance our understanding of blazar behavior across different activity
phases, especially for high-redshift blazars.

Similar to the local Universe, only a small fraction ($\sim 10\%$) of high-$z$
AGN have large radio loudness (Kratzer & Richards, 2015; Liu et al., 2021). In
particular, a limited number of radio-loud AGN with radio loudness $R>$10 have
been identified at $z>$5\. The properties of J1702+1301 are consistent with
those observed in other high-redshift blazars, such as J0309+2717 (Belladitta
et al. 2020; Spingola et al. 2020, $z=6.10$), Q0906+6930 (Romani et al. 2004,
$z=5.47$), and SDSS J102623.61+254259.5 (Sbarrato et al. 2012,$z=5.3$). These
blazars exhibit strong radio and X-ray emission, flat radio spectra, and radio
variability – typical features of blazars. In particular, both J0309+2717 and
Q0906+6930, observed with VLBI, show less powerful jets with lower proper
motion and bulk Lorentz factors compared to typical low-redshift blazars.
However, the high brightness temperature and rapid variability observed in
J1702+1301 suggest that it could be one of the most active blazars at
redshifts $>5.5$. This is in contrast to the less powerful jets observed in
the above blazars, suggesting a potentially unique jet behavior in J1702+1301.
The unique characteristics of J1702+1301, compared to other high-redshift
blazars, manifests the diversity in blazar properties and behaviors,
suggesting a more complex picture of AGN activity in the early stages of the
Universe.

Figure 2:  Radio spectral energy distribution of J1702+1301, including data
from our VLBA observation (red squares) as well as archive data from the GMRT,
ASKAP, and VLASS surveys (blue labels). The best fitting results of VLBA
observations is shown as the red line, with a spectral index of
$\alpha=-0.21\pm 0.001$. This spectrum is only slightly steeper than the
previous results (black line) fitted from the synthesis imaging with much
lower resolutions (An et al., 2023).

##  5 summary

The study of the SRGE J170245.3+130104, located at a redshift of $\sim$5.5,
represents a significant step in understanding high-redshift AGN and their jet
formations. Our observations with the VLBA have successfully detected
J1702+1301 across multiple frequencies, yet its jet structure remains
unresolved, highlighting the challenges in studying such distant objects.

The VLBA observations at 1.73 GHz, 2.26 GHz, and 4.87 GHz reveal a compact,
unresolved core, indicative of a highly active AGN. The spectral index derived
from these observations, $\alpha=-0.21$, is consistent with previous low-
resolution studies and supports the interpretation of a flat-spectrum radio
core. This finding, coupled with the high brightness temperatures observed,
reinforces J1702+1301’s classification as a blazar, particularly notable for
its rapid variability and exceptional brightness at such a high redshift. Our
findings, particularly the observations conducted during the blazar’s low flux
density state, provide a valuable baseline for understanding its typical
emission state. This contrasts with potential future observations during
higher activity periods, offering a dynamic view of the blazar’s radio
variability.

J1702+1301 contributes to the limited but growing sample of blazars known at
redshifts greater than 5, providing unique insights into the early stages of
jet formation and evolution in young AGN. Our study highlights the importance
of further high-resolution observations at varying frequencies to fully
understand the jet dynamics and emission characteristics of J1702+1301. Such
future observations are expected to offer more detailed information on the jet
properties, especially during periods of flaring activity. J1702+1301 shows
unique characteristics of rapid variability when compared to other $z>5$
blazars. This comparative analysis with other distant blazars indicates the
heterogeneity in their emission properties, challenging and enriching our
current understanding of blazars in the early Universe.

###### Acknowledgements.

This research has been supported by the National SKA Program of China
(2022SKA0120102, 2022SKA0130103). We gratefully acknowledge financial support
from Pinghu Laboratory. This project is funded from China Postdoctoral Science
Foundation (certification number: 2023M733625). YQL is supported by the
Shanghai Post-doctoral Excellence Program and Shanghai Sailing Program under
grant number 23YF1455700. SGG is supported by Youth Innovation Promotion
Association CAS Program under NO.2021258. ALW is thankful for the financial
support received from the University of Chinese Academy of Sciences,
appreciates the support and hospitality provide by the SKA Observatory and
Jodrell Bank Centre for Astrophysics at the University of Manchester. This
work used resources of China SKA Regional Centre prototype funded by the
Ministry of Science and Technology of the People’s Republic of China and the
Chinese Academy of Sciences. The National Radio Astronomy Observatory (NRAO)
is a facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc. This paper makes use of the VLBA
data from program BR240.

## References

  * An et al. (2022) An, T., Wu, X., Lao, B., et al. 2022, Science China Physics, Mechanics, and Astronomy, 65, 129501
  * An et al. (2019) An, T., Wu, X.-P., & Hong, X. 2019, Nature Astronomy, 3, 1030
  * An et al. (2023) An, T., Wang, A., Liu, Y., et al. 2023, MNRAS, 519, 4047
  * Belladitta et al. (2020) Belladitta, S., Moretti, A., Caccianiga, A., et al. 2020, A&A, 635, L7
  * Cao et al. (2014) Cao, H. M., Frey, S., Gurvits, L. I., et al. 2014, A&A, 563, A111
  * Cheng et al. (2020) Cheng, X. P., An, T., Frey, S., et al. 2020, ApJS, 247, 57
  * Condon et al. (1982) Condon, J. J., Condon, M. A., Gisler, G., & Puschell, J. J. 1982, ApJ, 252, 102
  * Fan et al. (2006) Fan, X., Strauss, M. A., Richards, G. T., et al. 2006, AJ, 131, 1203
  * Frey et al. (2011) Frey, S., Paragi, Z., Gurvits, L. I., Gabányi, K. É., & Cseh, D. 2011, A&A, 531, L5
  * Greisen (2003) Greisen, E. W. 2003, in Astrophysics and Space Science Library, Vol. 285, Information Handling in Astronomy - Historical Vistas, ed. A. Heck, 109
  * Khorunzhev et al. (2021) Khorunzhev, G. A., Meshcheryakov, A. V., Medvedev, P. S., et al. 2021, Astronomy Letters, 47, 123
  * Kratzer & Richards (2015) Kratzer, R. M., & Richards, G. T. 2015, AJ, 149, 61
  * Liu et al. (2021) Liu, Y., Wang, R., Momjian, E., et al. 2021, ApJ, 908, 124
  * Merloni et al. (2020) Merloni, A., Nandra, K., & Predehl, P. 2020, Nature Astronomy, 4, 634
  * Romani et al. (2004) Romani, R. W., Sowards-Emmerd, D., Greenhill, L., & Michelson, P. 2004, ApJ, 610, L9
  * Sbarrato et al. (2012) Sbarrato, T., Ghisellini, G., Nardini, M., et al. 2012, MNRAS, 426, L91
  * Shepherd (1997) Shepherd, M. C. 1997, in Astronomical Society of the Pacific Conference Series, Vol. 125, Astronomical Data Analysis Software and Systems VI, ed. G. Hunt & H. Payne, 77
  * Spingola et al. (2020) Spingola, C., Dallacasa, D., Belladitta, S., et al. 2020, A&A, 643, L12
  * Willott et al. (2010) Willott, C. J., Delorme, P., Reylé, C., et al. 2010, AJ, 139, 906
  * Yang et al. (2017) Yang, J., Fan, X., Wu, X.-B., et al. 2017, AJ, 153, 184
  * Yang et al. (2023) Yang, J., Fan, X., Gupta, A., et al. 2023, arXiv e-prints, arXiv:2302.01777
  * Zhang et al. (2022) Zhang, Y., An, T., Wang, A., et al. 2022, A&A, 662, L2


# Jones Polynomial versus Determinant of Quasi-Alternating Links

Khaled Qazaqzeh  Permanent Address: Department of Mathematics, Faculty of
Science, Yarmouk University, Irbed, Jordan, 21163<EMAIL_ADDRESS>https://faculty.yu.edu.jo/qazaqzeh Current Address: Department of Mathematics,
Faculty of Science, Kuwait University, P. O. Box 5969 Safat-13060, Kuwait,
State of Kuwait<EMAIL_ADDRESS>
(Date: 05/08/2022)

###### Abstract.

We prove that there are only finitely many values of the Jones polynomial of
quasi-alternating links of a given determinant. Consequently, we prove that
there are only finitely many quasi-alternating links of a given Jones
polynomial iff there are only finitely many quasi-alternating links of a given
determinant.

###### Key words and phrases:

quasi-alternating links, Jones polynomial, determinant

###### 2000 Mathematics Subject Classification:

57K10, 57K14

##  1\. introduction

The study of the class of alternating links has played a crucial role in the
advancement and evolution of knot theory in the last century. In particular,
some long lasting conjectures about alternating links have been solved using
the study of the Jones polynomial. Therefore, it is very natural to study the
Jones polynomial of any class of links in trying to get a better understanding
of the given class.

It is easy to see that there are only finitely many alternating links of a
given determinant. This is true as a result of the fact that there are only
finitely many links of a crossing number less than or equal to some positive
integer and the result due to Bankwitz in [1] that the determinant in the
class of alternating links represents an upper bound of the crossing number.
This implies that there are only finitely many values of the Jones polynomial
of alternating links of a given determinant. Moreover, we can conclude that
there are only finitely many alternating links of a given Jones polynomial as
a consequence of the previous result and the well-known fact that the
determinant is the absolute value of the evaluation of the Jones polynomial at
$t=-1$.

In trying to naturally generalize the class of alternating links, the authors
in [12] defined the class of quasi-alternating links. Contrary to alternating
links that admits a simple diagrammatic definition, this class has been
defined recursively as follows:

######  Definition 1.1.

The set $\mathcal{Q}$ of quasi-alternating links is the smallest set
satisfying the following properties:

  * •

The unknot belongs to $\mathcal{Q}$.

  * •

If $L$ is a link with a diagram $D$ containing a crossing $c$ such that

    1. (1)

both smoothings of the diagram $D$ at the crossing $c$, $L_{0}$ and $L_{1}$ as
in Figure 1 belong to $\mathcal{Q}$;

    2. (2)

$\det(L_{0}),\det(L_{1})\geq 1$;

    3. (3)

$\det(L)=\det(L_{0})+\det(L_{1})$; then $L$ is in $\mathcal{Q}$ and in this
case we say $L$ is quasi-alternating at the crossing $c$ with quasi-
alternating diagram $D$.

$L$ $L_{0}$ $L_{1}$

Figure 1. The link diagram $L$ at the crossing $c$ and its smoothings $L_{0}$
and $L_{1}$ respectively.

It is easy to apply this definition to determine if a given link diagram is
quasi-alternating and thus the link represented by this diagram is quasi-
alternating. But it can not be used to determine if a given link is not quasi-
alternating because of the unlimited number of link diagrams the represent the
same link. Rather than that, several obstructions for a link to be quasi-
alternating have been introduced through the past two decades. Many of these
have been established for the class of alternating links first, then
generalized later to quasi-alternating links. Some of these main obstructions
are listed here.

  1. (1)

the branched double-cover of any quasi-alternating link is an $L$-space [12,
Proposition. 3.3];

  2. (2)

the branched double-cover of any quasi-alternating link bounds a negative
definite $4$-manifold $W$ with $H_{1}(W)=0$ [12, Proof of Lemma. 3.6];

  3. (3)

the $\mathbb{Z}/2\mathbb{Z}$ knot Floer homology group of any quasi-
alternating link is $\sigma$-thin [10, Theorem. 2];

  4. (4)

the reduced ordinary Khovanov homology group of any quasi-alternating link is
$\sigma$-thin [10, Theorem. 1];

  5. (5)

the reduced odd Khovanov homology group of any quasi-alternating link is
$\sigma$-thin [11, Remark after Proposition. 5.2];

  6. (6)

the determinant of any quasi-alternating link is bigger than the degree of its
$Q$-polynomial [15, Theorem 2.2]. This inequality was later sharpened to the
determinant minus one is bigger than or equal to the degree of the
$Q$-polynomial with equality holds only for $(2,n)$-torus links [17, Theorem
1.1].

  7. (7)

The length of any gap in the differential grading of the Khovanov homology of
any quasi-alternating link is at most one [14, Theorem 2.5].

Two natural questions can be raised at this time in trying to generalize the
previous discussion to the class of quasi-alternating links as follows:

######  Question 1.2.

  1. (1)

Is it true that there are only finitely many values of the Jones polynomial of
quasi-alternating links of a given determinant?

  2. (2)

Is it true that there are only finitely many quasi-alternating links of a
given Jones polynomial?

These two question clearly have a positive answer if we assume that there are
only finitely many quasi-alternating links of a given determinant similar to
the case of alternating links. The positive answer for the second question
under this assumption follows directly as a consequence of the fact that the
determinant of any link is equal to the absolute value of the evaluation of
the Jones polynomial at $t=-1$.

The finiteness of quasi-alternating links of a given determinant was pointed
out for the first time in Conjecture 3.3 in [3]. This conjecture has been
verified for some values of the determinant in [3, 8, 18] and was later
generalized to a conjecture that states that the determinant is an upper bound
of the crossing number in the class of quasi-alternating links in [16,
Conjecture 1.1]. This later conjecture has been verified for several
subclasses of quasi-alternating links. These two conjectures are still open as
far as we know.

In this paper, we prove that the first question has a positive answer without
assuming the finiteness of quasi-alternating links of a given determinant.
Also as a direct consequence, we prove that the second question has a positive
answer iff the finiteness of quasi-alternating links of a given determinant
assumption holds. In particular, our main goal in this paper is to prove the
following theorem and give some applications.

######  Theorem 1.3.

There are only finitely many values of the Jones polynomial of quasi-
alternating links of a given determinant.

This paper is organized as follows. In Section 2, we recall the definition of
the Jones polynomial in terms of the Kauffman bracket and review some of its
properties that will be used in the next section. In Section 3, we give the
proof of our main result and give some applications and consequences.

##  2\. The Jones polynomial

All the results of this paper are restricted to the case where the link $L$ as
illustrated in Figure 1. Similar results can be obtained for the other case by
interchanging $A$ and $A^{-1}$ in the third formula in Definition 2.1.

The Jones polynomial $V_{L}(t)$ is an invariant of oriented links that was
defined for the first time in [4]. It is a Laurent polynomial with integral
coefficients that can be defined in several ways. In this subsection, we shall
briefly recall the definition of this polynomial in terms of the Kauffman
bracket as it has been defined in [6].

######  Definition 2.1.

The Kauffman bracket polynomial is a function from the set of unoriented link
diagrams in the oriented plane to the ring of Laurent polynomials with integer
coefficients in an indeterminate $A$. It maps a link $L$ to $\left\langle
L\right\rangle\in\mathbb{Z}[A^{-1},A]$ and it is defined by the following
relations:

  1. (1)

$\left\langle\bigcirc\right\rangle=1$,

  2. (2)

$\left\langle\bigcirc\cup L\right\rangle=(-A^{-2}-A^{2})\left\langle
L\right\rangle$,

  3. (3)

$\left\langle L\right\rangle=A\left\langle
L_{0}\right\rangle+A^{-1}\left\langle L_{1}\right\rangle$,

where $\bigcirc$, in relation 2 above, denotes a trivial circle disjoint from
the rest of the link, and $L,L_{0},\text{and }L_{1}$ represent three
unoriented links which are identical everywhere except in a small region where
they are as indicated in Figure 1.

For any oriented link diagram $L$, we let $x(L)$ denote the number of negative
crossings and $y(L)$ denote the number of positive crossings in $L$, see
Figure 2. Also, we define the writhe of $L$ to be the integer
$w(L)=y(L)-x(L)$. It is easy to see that the writhes of the links $L,L_{0}$,
and $L_{1}$ are related as follows:

######  Remark 2.2.

  1. (1)

If the crossing is positive, then we have
$x(L_{0})=x(L),y(L_{0})=y(L)-1,x(L_{1})=x(L)+e$ and $y(L_{1})=y(L)-e-1$.
Therefore, $w(L_{0})=w(L)-1$ and $w(L_{1})=w(L)-2e-1$, where $e$ denotes the
difference between the number of negative crossings in $L_{1}$ and the number
of negative crossings in $L$.

  2. (2)

If the crossing is negative, then we have
$x(L_{0})=x(L)-1,y(L_{0})=y(L),x(L_{1})=x(L)-e-1$ and $y(L_{1})=y(L)+e$.
Therefore, $w(L_{0})=w(L)+1$ and $w(L_{1})=w(L)+2e+1$, where $e$ denotes the
difference between the number of positive crossings in $L_{1}$ and the number
of positive crossings in $L$.

######  Definition 2.3.

The Jones polynomial $V_{L}(t)$ of an oriented link $L$ is the Laurent
polynomial in $t^{\pm 1/2}$ with integer coefficients defined by

$V_{L}(t)=((-A)^{-3w(L)}\left\langle
L\right\rangle)_{t^{1/2}=A^{-2}}\in\mathbb{Z}[t^{-1/2},t^{1/2}],$

where $\left\langle L\right\rangle$ is the Kauffman bracket of the unoriented
link obtained from $L$ by ignoring the orientation.

Figure 2. Negative and positive crossings respectively.

The following lemma is a direct consequence of Remark 2.2 and it gives the
relation between the Jones polynomial of the link and the Jones polynomial of
the links $L_{0}$ and $L_{1}$.

######  Lemma 2.4.

The Jones polynomial of the link $L$ at the crossing $c$ satisfies one of the
following skein relations:

  1. (1)

If $c$ is a positive crossing, then
$V_{L}(t)=-t^{\frac{1}{2}}V_{L_{0}}(t)-t^{\frac{3e}{2}+1}V_{L_{1}}(t)$.

  2. (2)

If $c$ is a negative crossing, then
$V_{L}(t)=-t^{\frac{3e}{2}-1}V_{L_{0}}(t)-t^{\frac{-1}{2}}V_{L_{1}}(t)$.

where $e$ has been defined in Remark 2.2.

We close this section by the following lemma that will be used in the proof of
the main theorem.

######  Lemma 2.5.

The absolute value of any coefficient of any quasi-alternating link is bounded
above by the determinant.

###### Proof.

The main result of [10] yields that the Jones polynomial of any quasi-
alternating link is alternating. In other words, if
$V_{L}(t)=t^{m}\sum_{i=0}^{n}a_{i}t^{i}$ then $a_{i}a_{i+1}\leq 0$ for $0\leq
i\leq n-1$. Now the result follows easily since the determinant of such a link
is equal to the absolute value of the alternating sum of the coefficients of
the Jones polynomial. ∎

##  3\. Proof of the Main Theorem and Some Consequences

We start this section by introducing a new terminology to simplify notation
for this section. We can introduce an equivalence relation on the class of
quasi-alternating links for any link invariant according to the following
definition.

######  Definition 3.1.

For any link invariant, we define a relation on the class of quasi-alternating
links by two quasi-alternating links are related in this invariant relation if
they have the same value of this link invariant.

######  Remark 3.2.

It is easy to see that such a relation is an equivalence relation for any link
invariant and its equivalence classes will be called by the given link
invariant.

###### Proof of Theorem 1.3.

We apply induction on the determinant of the given quasi-alternating link to
prove the statement of the theorem. If $\det(L)=1$, then the result holds
since the only quasi-alternating link of determinant one is the unknot
according to the definition.

Now, suppose that the result holds when the determinant is less than or equal
to $n$. In other words, there is a finite list of Jones polynomial equivalence
classes of quasi-alternating links of determinant less than or equal to $n$.
In particular, the list $\\{K_{1},K_{2},\ldots,K_{m}\\}$ consists of all Jones
polynomial equivalence classes of quasi-alternating links of determinant less
than or equal to $n$. In particular, if $K$ is a quasi-alternating link of
determinant less than or equal to $n$, then $V_{K}(t)=V_{K_{i}}(t)$ for some
$1\leq i\leq m$.

The definition of $L$ being quasi-alternating of determinant $n+1$ implies
that the links $L_{0}$ and $L_{1}$ have determinant less than $n+1$. By a
direct application of the induction hypothesis on the links $L_{0}$ and
$L_{1}$, we have only finitely many values of $V_{L_{0}}(t)$ and
$V_{L_{1}}(t)$ since $L_{0}$ and $L_{1}$ represent two Jones polynomial
equivalence classes in the above list.

Now the Jones polynomial of the link $L$ is given by one of the two formulas
in Lemma 2.4 in terms of the Jones polynomial of the links $L_{0}$ and
$L_{1}$. The choice of the formula depends on the sign of the given crossing.

We let $m=\min\\{\operatorname{mindeg}V_{K_{i}}(t)\ |\ 1\leq i\leq n\\}$ and
$M=\max\\{\operatorname{maxdeg}V_{K_{i}}(t)\ |\ 1\leq i\leq n\\}$ for the two
cases of the proof according to Lemma 2.4:

  1. (1)

For the first case, we have
$V_{L}(t)=-t^{\frac{1}{2}}V_{L_{0}}(t)-t^{\frac{3e}{2}+1}V_{L_{1}}(t)$ with
only finitely many values of the number $e$. Otherwise this will force
consecutive coefficients of $V_{L}(t)$ to vanish and this contradicts with the
result of [14, Corollary 4.1]. This argument implies that there will be only
finitely many values of $V_{L}(t)$ since $\operatorname{mindeg}V_{L}(t)\geq
m-\left|\frac{3e}{2}+1\right|$ and $\operatorname{maxdeg}V_{L}(t)\leq
M+\left|\frac{3e}{2}+1\right|$ and the absolute value of the coefficients of
$V_{L}(t)$ are bounded above by $n+1$ as a result of Lemma 2.5.

  2. (2)

For the second case, we have
$V_{L}(t)=-t^{\frac{3e}{2}-1}V_{L_{0}}(t)-t^{\frac{-1}{2}}V_{L_{1}}(t)$ with
only finitely many values of the number $e$. Otherwise this will force
consecutive coefficients of $V_{L}(t)$ to vanish and this contradicts with the
result of [14, Corollary 4.1]. This argument implies that there will be only
finitely many values of $V_{L}(t)$ since $\operatorname{mindeg}V_{L}(t)\geq
m-\left|\frac{3e}{2}-1\right|$ and $\operatorname{maxdeg}V_{L}(t)\leq
M+\left|\frac{3e}{2}-1\right|$ and the absolute value of the coefficients of
$V_{L}(t)$ are bounded above by $n+1$ as a result of Lemma 2.5.

∎

######  Corollary 3.3.

There are only finitely many Jones polynomial equivalence classes of quasi-
alternating links of determinant less than or equal to any positive number.
Moreover, we conclude that there are only finitely many Khovanov homology
equivalence classes of quasi-alternating links of determinant less than or
equal to any positive number.

######  Remark 3.4.

We can conclude that the breadth of the Jones polynomial of any quasi-
alternating link in a determinant equivalence class is bounded above by some
positive integer. This upper bound clearly depends entirely on the determinant
of the equivalence class and it is conjectured to be the determinant itself in
[13, Conjecture 3.8].

######  Corollary 3.5.

In any infinite family of links of the same determinant, either this family
has only finitely many values of the Jones polynomial or it has only finitely
many quasi-alternating links.

######  Example 3.6.

Kanenobu in [5] constructed an infinite family of knots of the same
determinant that have infinitely many values of the Jones polynomial.
Therefore, we can conclude that this family contains only finitely many quasi-
alternating links which confirms the result of [15, Corollary 3.3]. Moreover,
this implies that the result of Theorem 1.3 does not hold in the class of all
links.

One might wonder if the Jones polynomial equivalence classes are finite in the
class of quasi-alternating links. It is easy to see that some of the Jones
polynomial equivalence classes are finite. For example, the Jones polynomial
equivalence classes of
$1,-t^{\frac{-1}{2}}-t^{\frac{-5}{2}},-t^{-4}+t^{-3}+t^{-1},-t^{\frac{-9}{2}}-t^{\frac{-5}{2}}+t^{\frac{-3}{2}}-t^{\frac{-1}{2}},t^{-2}-t^{-1}+1-t+t^{2},-t^{-7}+t^{-6}-t^{-5}+t^{-4}+t^{-2},-t^{\frac{-17}{2}}+t^{\frac{-15}{2}}-t^{\frac{-13}{2}}+t^{\frac{-11}{2}}-t^{\frac{-9}{2}}-t^{\frac{-5}{2}},-t^{-10}+t^{-9}-t^{-8}+t^{-7}-t^{-6}+t^{-5}+t^{-3}$
and $-t^{-6}+t^{-5}-t^{-4}+2t^{-3}-t^{-2}+t^{-1}$ are all finite since there
are only finitely many quasi-alternating links of determinant $1,2,3,4,5,$ and
$7$, respectively according to the results in [3, 8, 18].

As a consequence of the main result and based on the previous discussion, a
partial answer to Question 1.2(2) can be given in the following corollary:

######  Corollary 3.7.

In the class of quasi-alternating links, every determinant equivalence class
is finite iff every Jones polynomial equivalence class is finite.

###### Proof.

The first implication follows directly as a result of the fact that every
Jones polynomial equivalence class consists of links of the same determinant.
Now the second implication follows since each determinant equivalence class is
a finite union of Jones polynomial equivalence classes as a result of having
only finitely many values of the Jones polynomial of quasi-alternating links
of a given determinant. ∎

######  Remark 3.8.

The result in Corollary 3.7 still holds if Jones polynomial is replaced by
Khovanov homology based on the result of Corollary 3.3.

Moreover, we can generalize Question 1.2(2) to any other link invariant that
determines the determinant as follows:

######  Question 3.9.

In the class of quasi-alternating links and any link invariant that determines
the determinant, is it true that all invariant equivalence classes are finite?

The next result uses the twisting technique that was first introduced by
Champanerkar and Kofman [2, Page 2452]. It basically replaces the crossing $c$
of the link $L$ by some alternating integer tangle. The original technique
uses not only integer tangle but also a rational tangle and it is applied only
at the crossing where the given link is quasi-alternating. Now we can exhibit
an infinite family of non quasi-alternating links from a given particular link
using the twisting technique as follows:

######  Corollary 3.10.

Let $L^{n}$ be the link obtained by replacing the crossing $c$ of the link $L$
by an alternating integer tangle of $n$ crossings such that the type of each
crossing is of the same type as the type of $c$. If $\det(L_{0})=0$ or
$\det(L_{1})=0$ and $V_{L_{0}}(t)$ and $V_{L_{1}}(t)$ are nonzero, then
$\\{L^{n}|n\in\mathbb{N}\\}$ contains only finitely many quasi-alternating
links.

###### Proof.

The result follows since $\det(L^{n})=\det(L)$ for any $n$ and
$V_{L^{i}}(t)\neq V_{L^{j}}(t)$ for $i\neq j$ according to the formulas in
Lemma 2.4. This infinite family of links consists of links of the same
determinant with infinitely many values of the Jones polynomial. Thus, only
finitely many links of this family are quasi-alternating. ∎

As a direct application of the above corollary, we give the following example:

######  Example 3.11.

Let $K_{l}^{m}=K_{\beta}(\sigma^{2l},\sigma^{-2m})$ with
$\beta=\sigma_{1}^{-1}\sigma_{2}\sigma^{-2n}$ be the knot defined in [7] and
used later in [19] for any positive integers $l$ and $m$. It is not too hard
to prove that $\det(K_{l}^{m})=(\det(\beta))^{2}$ and $V_{K_{l}^{m}}(t)\neq
V_{K_{l^{{}^{\prime}}}^{m}}(t)$ for $l\neq l^{{}^{\prime}}$. Then according to
the result of Corollary 3.10, the family of knots $\\{K_{l}^{m}|\text{for
fixed}\ m\\}$ contains only finitely many quasi-alternating links.

## References

  * [1] C. Bankwitz, $\ddot{\text{U}}$ber die Torsionszahlen der alternierenden Knoten, Mathematische Annalen, 103(1): 145–161, 1930.
  * [2] A. Champanerkar and I. Kofman, Twisting quasi-alternating links, Proc. Amer. Math. Soc., 137(7): 2451–2458, 2009.
  * [3] J. Greene,  Homologically thin, non-quasi-alternating links, Math. Res. Lett., 17(1): 39–49, 2010.
  * [4] V. Jones, A Polynomial invariant for knots via von Neumann algebras, Bull. Am. Math. Soc. 12: 103–111, 1985.
  * [5] T. Kanenobu, Infinitely many knots with the same polynomial invariant, Proc. Amer. Math. Soc. 97 (1) (1986), 158–162.
  * [6] L. Kauffman, State models and the Jones polynomial, Topology 26 (3) (1987): 395–407.
  * [7] K. Luse and Y. Rong, Examples of knots with the same polynomial, J. Knot Theory Ramif. 15 (6): 749-759, 2006.
  * [8] T. Lidman, and S. Sivek, Quasi-alternating links with small determinant, Math. Proc. Cambridge Philos. Soc., 162 (2): 319–336, 2017.
  * [9] C. Manolescu, An unoriented skein exact triangle for knot Floer homology, Math. Res. Lett. 14 (5): 839–852, 2007.
  * [10] C. Manolescu and P. Ozsv$\acute{\text{a}}$th, On the Khovanov and knot Floer homologies of quasi-alternating links. In Proceedings of G$\ddot{\text{o}}$kova Geometry-Topology Conference (2007), 60-81. G$\ddot{\text{o}}$kova Geometry/Topology Conference (GGT), G$\ddot{\text{o}}$kova, 2008.
  * [11] P. Ozsv$\acute{\text{a}}$th, J. Rasmussen and Z. Szab$\acute{\text{o}}$, Odd Khovanov homology, Alg. Geom. Top. 13(3): 1465-1488, 2013.
  * [12] P. Ozsv$\acute{a}$th and Z. Szab$\acute{\text{o}}$, On the Heegaard Floer homology of branched double-covers, Adv. Math., 194 (1): 1–33, 2005.
  * [13] K. Qazaqzeh and N. Chbili, A new obstruction of quasi-alternating links, Algebr. Geom. Topol. 15 (2015) 1847-1862.
  * [14] K. Qazaqzeh, and N. Chbili, On Khovanov homology of quasi-alternating links, Mediterr. J. Math., 19 (3) (2022). https://doi.org/10.1007/s00009-022-02006-5.
  * [15] K. Qazaqzeh and N. Chbili, A new obstruction of quasi-alternating links, Algebr. Geom. Topol. 15: 1847–1862, 2015.
  * [16] K. Qazaqzeh, B. Qublan, and A. Jaradat, A remark on the determinant of quasi-alternating links, J. Knot Theory Ramif. 22 (6): 1350031, 2013.
  * [17] M. Teragaito, Quasi-alternating links and $Q$-polynomials, J. Knot Theory Ramif. 23 (12), 1450068, 2014.
  * [18] M. Teragaito, Quasi-alternating links and Kauffman polynomials, J. Knot Theory Ramif. 24 (7), 1550038, 2015.
  * [19] L. Watson, Knots with identical Khovanov homology, Algebr. Geom. Topol. 7: 1389-1407, 2007.