9 39

9_38

9_40

Visit 9 39's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 39's page at Knotilus! Visit 9 39's page at the original Knot Atlas! 9 39 Quick Notes

9 39 Further Notes and Views

Knot presentations

Planar diagram presentation	X1627 X3,11,4,10 X7,18,8,1 X17,13,18,12 X9,17,10,16 X5,15,6,14 X15,5,16,4 X11,3,12,2 X13,9,14,8
Gauss code	-1, 8, -2, 7, -6, 1, -3, 9, -5, 2, -8, 4, -9, 6, -7, 5, -4, 3
Dowker-Thistlethwaite code	6 10 14 18 16 2 8 4 12
Conway Notation	[2:2:20]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-1][-10] Hyperbolic Volume 12.8103 A-Polynomial See Data:9 39/A-polynomial

[edit Notes for 9 39's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 2

[edit Notes for 9 39's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{2}+14t-21+14t^{-1}-3t^{-2}$
Conway polynomial	$-3z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 55, 2 }
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+8q^{5}-9q^{4}+10q^{3}-8q^{2}+6q-3+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-2z^{4}a^{-4}+z^{2}a^{-2}-3z^{2}a^{-4}+3z^{2}a^{-6}+z^{2}+2a^{-2}-2a^{-4}+2a^{-6}-a^{-8}$
Kauffman polynomial (db, data sources)	$2z^{8}a^{-4}+2z^{8}a^{-6}+5z^{7}a^{-3}+9z^{7}a^{-5}+4z^{7}a^{-7}+5z^{6}a^{-2}+5z^{6}a^{-4}+3z^{6}a^{-6}+3z^{6}a^{-8}+3z^{5}a^{-1}-7z^{5}a^{-3}-18z^{5}a^{-5}-7z^{5}a^{-7}+z^{5}a^{-9}-7z^{4}a^{-2}-15z^{4}a^{-4}-13z^{4}a^{-6}-6z^{4}a^{-8}+z^{4}-3z^{3}a^{-1}+5z^{3}a^{-3}+12z^{3}a^{-5}+2z^{3}a^{-7}-2z^{3}a^{-9}+5z^{2}a^{-2}+12z^{2}a^{-4}+9z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}-za^{-3}-3za^{-5}-za^{-7}+za^{-9}-2a^{-2}-2a^{-4}-2a^{-6}-a^{-8}$
The A2 invariant	$q^{4}-q^{2}-1+3q^{-2}-q^{-4}+2q^{-6}+q^{-8}-q^{-10}+q^{-12}-2q^{-14}+2q^{-16}-q^{-20}+2q^{-22}-q^{-24}-q^{-26}$
The G2 invariant	$q^{18}-2q^{16}+4q^{14}-6q^{12}+5q^{10}-3q^{8}-2q^{6}+12q^{4}-19q^{2}+28-30q^{-2}+21q^{-4}-3q^{-6}-27q^{-8}+58q^{-10}-76q^{-12}+73q^{-14}-45q^{-16}-6q^{-18}+63q^{-20}-97q^{-22}+101q^{-24}-61q^{-26}+2q^{-28}+53q^{-30}-80q^{-32}+65q^{-34}-12q^{-36}-45q^{-38}+87q^{-40}-83q^{-42}+36q^{-44}+37q^{-46}-103q^{-48}+134q^{-50}-123q^{-52}+66q^{-54}+10q^{-56}-84q^{-58}+131q^{-60}-134q^{-62}+95q^{-64}-29q^{-66}-43q^{-68}+87q^{-70}-93q^{-72}+59q^{-74}-52q^{-78}+80q^{-80}-61q^{-82}+8q^{-84}+57q^{-86}-100q^{-88}+103q^{-90}-65q^{-92}-q^{-94}+60q^{-96}-93q^{-98}+95q^{-100}-63q^{-102}+19q^{-104}+19q^{-106}-45q^{-108}+45q^{-110}-33q^{-112}+17q^{-114}-3q^{-116}-6q^{-118}+8q^{-120}-8q^{-122}+5q^{-124}-2q^{-126}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 9_39.

A1 Invariants.

Weight	Invariant
1	$q^{3}-2q+3q^{-1}-2q^{-3}+2q^{-5}+q^{-7}-q^{-9}+2q^{-11}-3q^{-13}+2q^{-15}-q^{-17}$
2	$q^{10}-2q^{8}+6q^{4}-8q^{2}-3+18q^{-2}-10q^{-4}-13q^{-6}+21q^{-8}-2q^{-10}-15q^{-12}+11q^{-14}+7q^{-16}-7q^{-18}-6q^{-20}+11q^{-22}+2q^{-24}-18q^{-26}+10q^{-28}+12q^{-30}-20q^{-32}+2q^{-34}+16q^{-36}-11q^{-38}-5q^{-40}+8q^{-42}-q^{-44}-2q^{-46}+q^{-48}$
3	$q^{21}-2q^{19}+3q^{15}-7q^{11}-q^{9}+19q^{7}+4q^{5}-35q^{3}-18q+50q^{-1}+49q^{-3}-58q^{-5}-81q^{-7}+48q^{-9}+110q^{-11}-22q^{-13}-124q^{-15}-8q^{-17}+120q^{-19}+35q^{-21}-92q^{-23}-56q^{-25}+63q^{-27}+66q^{-29}-28q^{-31}-69q^{-33}-7q^{-35}+70q^{-37}+34q^{-39}-68q^{-41}-65q^{-43}+66q^{-45}+87q^{-47}-51q^{-49}-109q^{-51}+28q^{-53}+119q^{-55}+3q^{-57}-116q^{-59}-33q^{-61}+92q^{-63}+58q^{-65}-59q^{-67}-66q^{-69}+28q^{-71}+54q^{-73}-35q^{-77}-11q^{-79}+17q^{-81}+8q^{-83}-5q^{-85}-4q^{-87}+q^{-89}+2q^{-91}-q^{-93}$
4	$q^{36}-2q^{34}+3q^{30}-3q^{28}+q^{26}-5q^{24}+7q^{22}+16q^{20}-16q^{18}-18q^{16}-29q^{14}+34q^{12}+94q^{10}+4q^{8}-88q^{6}-177q^{4}-4q^{2}+269+223q^{-2}-46q^{-4}-458q^{-6}-324q^{-8}+281q^{-10}+587q^{-12}+333q^{-14}-510q^{-16}-757q^{-18}-93q^{-20}+656q^{-22}+779q^{-24}-153q^{-26}-823q^{-28}-512q^{-30}+308q^{-32}+823q^{-34}+256q^{-36}-471q^{-38}-598q^{-40}-94q^{-42}+514q^{-44}+420q^{-46}-67q^{-48}-446q^{-50}-319q^{-52}+170q^{-54}+444q^{-56}+222q^{-58}-308q^{-60}-471q^{-62}-86q^{-64}+492q^{-66}+480q^{-68}-189q^{-70}-631q^{-72}-374q^{-74}+475q^{-76}+743q^{-78}+71q^{-80}-654q^{-82}-709q^{-84}+199q^{-86}+807q^{-88}+460q^{-90}-333q^{-92}-826q^{-94}-250q^{-96}+475q^{-98}+623q^{-100}+160q^{-102}-511q^{-104}-453q^{-106}-18q^{-108}+365q^{-110}+359q^{-112}-71q^{-114}-252q^{-116}-199q^{-118}+31q^{-120}+191q^{-122}+81q^{-124}-19q^{-126}-92q^{-128}-50q^{-130}+31q^{-132}+27q^{-134}+19q^{-136}-11q^{-138}-14q^{-140}+2q^{-142}+q^{-144}+4q^{-146}-q^{-148}-2q^{-150}+q^{-152}$
5	$q^{55}-2q^{53}+3q^{49}-3q^{47}-2q^{45}+3q^{43}+3q^{41}+4q^{39}+3q^{37}-16q^{35}-28q^{33}+q^{31}+45q^{29}+67q^{27}+26q^{25}-78q^{23}-176q^{21}-133q^{19}+107q^{17}+362q^{15}+373q^{13}-4q^{11}-574q^{9}-844q^{7}-376q^{5}+690q^{3}+1486q+1143q^{-1}-420q^{-3}-2090q^{-5}-2322q^{-7}-436q^{-9}+2359q^{-11}+3622q^{-13}+1876q^{-15}-1907q^{-17}-4647q^{-19}-3684q^{-21}+702q^{-23}+4997q^{-25}+5332q^{-27}+1068q^{-29}-4429q^{-31}-6381q^{-33}-2940q^{-35}+3094q^{-37}+6539q^{-39}+4408q^{-41}-1367q^{-43}-5793q^{-45}-5162q^{-47}-316q^{-49}+4443q^{-51}+5172q^{-53}+1581q^{-55}-2899q^{-57}-4539q^{-59}-2348q^{-61}+1461q^{-63}+3655q^{-65}+2656q^{-67}-365q^{-69}-2756q^{-71}-2693q^{-73}-426q^{-75}+2053q^{-77}+2705q^{-79}+986q^{-81}-1633q^{-83}-2831q^{-85}-1472q^{-87}+1364q^{-89}+3173q^{-91}+2078q^{-93}-1183q^{-95}-3652q^{-97}-2847q^{-99}+821q^{-101}+4136q^{-103}+3841q^{-105}-172q^{-107}-4404q^{-109}-4877q^{-111}-889q^{-113}+4209q^{-115}+5774q^{-117}+2249q^{-119}-3374q^{-121}-6210q^{-123}-3713q^{-125}+1949q^{-127}+5923q^{-129}+4882q^{-131}-119q^{-133}-4818q^{-135}-5431q^{-137}-1681q^{-139}+3105q^{-141}+5089q^{-143}+2990q^{-145}-1136q^{-147}-3955q^{-149}-3519q^{-151}-540q^{-153}+2391q^{-155}+3163q^{-157}+1561q^{-159}-845q^{-161}-2237q^{-163}-1830q^{-165}-226q^{-167}+1171q^{-169}+1467q^{-171}+718q^{-173}-315q^{-175}-884q^{-177}-723q^{-179}-124q^{-181}+366q^{-183}+470q^{-185}+238q^{-187}-59q^{-189}-223q^{-191}-183q^{-193}-32q^{-195}+70q^{-197}+84q^{-199}+39q^{-201}-7q^{-203}-33q^{-205}-22q^{-207}+3q^{-209}+8q^{-211}+4q^{-213}+2q^{-215}-q^{-217}-4q^{-219}+q^{-221}+2q^{-223}-q^{-225}$

A2 Invariants.

Weight	Invariant
1,0	$q^{4}-q^{2}-1+3q^{-2}-q^{-4}+2q^{-6}+q^{-8}-q^{-10}+q^{-12}-2q^{-14}+2q^{-16}-q^{-20}+2q^{-22}-q^{-24}-q^{-26}$
1,1	$q^{12}-4q^{10}+10q^{8}-20q^{6}+38q^{4}-62q^{2}+98-150q^{-2}+211q^{-4}-270q^{-6}+334q^{-8}-374q^{-10}+372q^{-12}-316q^{-14}+212q^{-16}-54q^{-18}-136q^{-20}+344q^{-22}-522q^{-24}+662q^{-26}-745q^{-28}+754q^{-30}-702q^{-32}+582q^{-34}-415q^{-36}+218q^{-38}-18q^{-40}-158q^{-42}+298q^{-44}-386q^{-46}+410q^{-48}-380q^{-50}+319q^{-52}-248q^{-54}+168q^{-56}-102q^{-58}+58q^{-60}-28q^{-62}+12q^{-64}-4q^{-66}+q^{-68}$
2,0	$q^{12}-q^{10}-2q^{8}+2q^{6}+5q^{4}-q^{2}-10+14q^{-4}-11q^{-8}+3q^{-10}+11q^{-12}-q^{-14}-10q^{-16}+3q^{-18}+6q^{-20}-3q^{-22}+2q^{-24}+3q^{-26}-2q^{-28}+8q^{-32}-5q^{-34}-10q^{-36}+2q^{-38}+9q^{-40}-4q^{-42}-12q^{-44}+6q^{-46}+9q^{-48}-2q^{-50}-8q^{-52}+6q^{-56}-3q^{-60}-q^{-62}+q^{-64}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{8}-2q^{6}+5q^{2}-6-q^{-2}+13q^{-4}-10q^{-6}-3q^{-8}+17q^{-10}-10q^{-12}-4q^{-14}+12q^{-16}-4q^{-18}-3q^{-20}+2q^{-22}+5q^{-24}-2q^{-26}-8q^{-28}+8q^{-30}+4q^{-32}-15q^{-34}+9q^{-36}+7q^{-38}-15q^{-40}+7q^{-42}+3q^{-44}-8q^{-46}+4q^{-48}+q^{-50}-2q^{-52}+q^{-54}$
1,0,0	$q^{5}-q^{3}-q^{-1}+3q^{-3}-q^{-5}+3q^{-7}+q^{-9}+q^{-11}-q^{-13}-q^{-15}-2q^{-19}+2q^{-21}+2q^{-25}-q^{-27}+2q^{-29}-q^{-31}-q^{-33}-q^{-35}$

B2 Invariants.

Weight	Invariant
0,1	$q^{8}-2q^{6}+4q^{4}-7q^{2}+10-13q^{-2}+17q^{-4}-16q^{-6}+15q^{-8}-9q^{-10}+4q^{-12}+4q^{-14}-12q^{-16}+20q^{-18}-27q^{-20}+30q^{-22}-31q^{-24}+28q^{-26}-22q^{-28}+16q^{-30}-6q^{-32}-q^{-34}+9q^{-36}-13q^{-38}+15q^{-40}-17q^{-42}+15q^{-44}-12q^{-46}+8q^{-48}-5q^{-50}+2q^{-52}-q^{-54}$
1,0	$q^{14}-2q^{10}-2q^{8}+2q^{6}+6q^{4}+q^{2}-8-7q^{-2}+6q^{-4}+15q^{-6}+q^{-8}-16q^{-10}-9q^{-12}+13q^{-14}+15q^{-16}-5q^{-18}-16q^{-20}+15q^{-24}+5q^{-26}-12q^{-28}-6q^{-30}+10q^{-32}+9q^{-34}-6q^{-36}-10q^{-38}+4q^{-40}+11q^{-42}-2q^{-44}-12q^{-46}-2q^{-48}+13q^{-50}+6q^{-52}-13q^{-54}-14q^{-56}+9q^{-58}+18q^{-60}-q^{-62}-17q^{-64}-8q^{-66}+12q^{-68}+11q^{-70}-5q^{-72}-10q^{-74}-q^{-76}+6q^{-78}+3q^{-80}-2q^{-82}-2q^{-84}+q^{-88}$

G2 Invariants.

Weight

Invariant

1,0

$q^{18}-2q^{16}+4q^{14}-6q^{12}+5q^{10}-3q^{8}-2q^{6}+12q^{4}-19q^{2}+28-30q^{-2}+21q^{-4}-3q^{-6}-27q^{-8}+58q^{-10}-76q^{-12}+73q^{-14}-45q^{-16}-6q^{-18}+63q^{-20}-97q^{-22}+101q^{-24}-61q^{-26}+2q^{-28}+53q^{-30}-80q^{-32}+65q^{-34}-12q^{-36}-45q^{-38}+87q^{-40}-83q^{-42}+36q^{-44}+37q^{-46}-103q^{-48}+134q^{-50}-123q^{-52}+66q^{-54}+10q^{-56}-84q^{-58}+131q^{-60}-134q^{-62}+95q^{-64}-29q^{-66}-43q^{-68}+87q^{-70}-93q^{-72}+59q^{-74}-52q^{-78}+80q^{-80}-61q^{-82}+8q^{-84}+57q^{-86}-100q^{-88}+103q^{-90}-65q^{-92}-q^{-94}+60q^{-96}-93q^{-98}+95q^{-100}-63q^{-102}+19q^{-104}+19q^{-106}-45q^{-108}+45q^{-110}-33q^{-112}+17q^{-114}-3q^{-116}-6q^{-118}+8q^{-120}-8q^{-122}+5q^{-124}-2q^{-126}+q^{-128}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 39"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-3t^{2}+14t-21+14t^{-1}-3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-3z^{4}+2z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 55, 2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{8}+3q^{7}-6q^{6}+8q^{5}-9q^{4}+10q^{3}-8q^{2}+6q-3+q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{4}a^{-2}-2z^{4}a^{-4}+z^{2}a^{-2}-3z^{2}a^{-4}+3z^{2}a^{-6}+z^{2}+2a^{-2}-2a^{-4}+2a^{-6}-a^{-8}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$2z^{8}a^{-4}+2z^{8}a^{-6}+5z^{7}a^{-3}+9z^{7}a^{-5}+4z^{7}a^{-7}+5z^{6}a^{-2}+5z^{6}a^{-4}+3z^{6}a^{-6}+3z^{6}a^{-8}+3z^{5}a^{-1}-7z^{5}a^{-3}-18z^{5}a^{-5}-7z^{5}a^{-7}+z^{5}a^{-9}-7z^{4}a^{-2}-15z^{4}a^{-4}-13z^{4}a^{-6}-6z^{4}a^{-8}+z^{4}-3z^{3}a^{-1}+5z^{3}a^{-3}+12z^{3}a^{-5}+2z^{3}a^{-7}-2z^{3}a^{-9}+5z^{2}a^{-2}+12z^{2}a^{-4}+9z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}-za^{-3}-3za^{-5}-za^{-7}+za^{-9}-2a^{-2}-2a^{-4}-2a^{-6}-a^{-8}$

Vassiliev invariants

V2 and V3:

(2, 4)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $8$ $32$ $32$ ${\frac {460}{3}}$ ${\frac {116}{3}}$ $256$ ${\frac {2144}{3}}$ ${\frac {320}{3}}$ $192$ ${\frac {256}{3}}$ $512$ ${\frac {3680}{3}}$ ${\frac {928}{3}}$ ${\frac {49111}{15}}$ $-{\frac {8524}{15}}$ ${\frac {97084}{45}}$ ${\frac {857}{9}}$ ${\frac {5191}{15}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$2 is the signature of 9 39. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 χ 17                   1 -1 15                 2   2 13               4 1   -3 11             4 2     2 9           5 4       -1 7         5 4         1 5       3 5           2 3     3 5             -2 1   1 4               3 -1   2                 -2 -3 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 39]]
Out[2]=	9
In[3]:=	PD[Knot[9, 39]]
Out[3]=	PD[X[1, 6, 2, 7], X[3, 11, 4, 10], X[7, 18, 8, 1], X[17, 13, 18, 12], X[9, 17, 10, 16], X[5, 15, 6, 14], X[15, 5, 16, 4], X[11, 3, 12, 2], X[13, 9, 14, 8]]
In[4]:=	GaussCode[Knot[9, 39]]
Out[4]=	GaussCode[-1, 8, -2, 7, -6, 1, -3, 9, -5, 2, -8, 4, -9, 6, -7, 5, -4, 3]
In[5]:=	BR[Knot[9, 39]]
Out[5]=	BR[5, {1, 1, 2, -1, -3, -2, 1, 4, 3, -2, 3, 4}]
In[6]:=	alex = Alexander[Knot[9, 39]][t]
Out[6]=	3 14 2 -21 - -- + -- + 14 t - 3 t 2 t t
In[7]:=	Conway[Knot[9, 39]][z]
Out[7]=	2 4 1 + 2 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 39], Knot[11, NonAlternating, 162]}
In[9]:=	{KnotDet[Knot[9, 39]], KnotSignature[Knot[9, 39]]}
Out[9]=	{55, 2}
In[10]:=	J=Jones[Knot[9, 39]][q]
Out[10]=	1 2 3 4 5 6 7 8 -3 + - + 6 q - 8 q + 10 q - 9 q + 8 q - 6 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 39], Knot[11, NonAlternating, 11], Knot[11, NonAlternating, 112]}
In[12]:=	A2Invariant[Knot[9, 39]][q]
Out[12]=	-4 -2 2 4 6 8 10 12 14 16 -1 + q - q + 3 q - q + 2 q + q - q + q - 2 q + 2 q - 20 22 24 26 q + 2 q - q - q
In[13]:=	Kauffman[Knot[9, 39]][a, z]
Out[13]=	2 2 2 -8 2 2 2 z z 3 z z 2 3 z 9 z 12 z -a - -- - -- - -- + -- - -- - --- - -- - z + ---- + ---- + ----- + 6 4 2 9 7 5 3 8 6 4 a a a a a a a a a a 2 3 3 3 3 3 4 4 5 z 2 z 2 z 12 z 5 z 3 z 4 6 z 13 z ---- - ---- + ---- + ----- + ---- - ---- + z - ---- - ----- - 2 9 7 5 3 a 8 6 a a a a a a a 4 4 5 5 5 5 5 6 6 6 15 z 7 z z 7 z 18 z 7 z 3 z 3 z 3 z 5 z ----- - ---- + -- - ---- - ----- - ---- + ---- + ---- + ---- + ---- + 4 2 9 7 5 3 a 8 6 4 a a a a a a a a a 6 7 7 7 8 8 5 z 4 z 9 z 5 z 2 z 2 z ---- + ---- + ---- + ---- + ---- + ---- 2 7 5 3 6 4 a a a a a a
In[14]:=	{Vassiliev[2][Knot[9, 39]], Vassiliev[3][Knot[9, 39]]}
Out[14]=	{0, 4}
In[15]:=	Kh[Knot[9, 39]][q, t]
Out[15]=	3 1 2 q 3 5 5 2 7 2 4 q + 3 q + ----- + --- + - + 5 q t + 3 q t + 5 q t + 5 q t + 3 2 q t t q t 7 3 9 3 9 4 11 4 11 5 13 5 4 q t + 5 q t + 4 q t + 4 q t + 2 q t + 4 q t + 13 6 15 6 17 7 q t + 2 q t + q t