K11a164

K11a163

K11a165

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a164 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X18,6,19,5 X14,8,15,7 X16,9,17,10 X2,11,3,12 X20,13,21,14 X22,16,1,15 X12,17,13,18 X6,20,7,19 X8,21,9,22
Gauss code	1, -6, 2, -1, 3, -10, 4, -11, 5, -2, 6, -9, 7, -4, 8, -5, 9, -3, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 10 18 14 16 2 20 22 12 6 8

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a164/ThurstonBennequinNumber Hyperbolic Volume 18.3585 A-Polynomial See Data:K11a164/A-polynomial

[edit Notes for K11a164's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[0,4]$ Rasmussen s-Invariant 0

[edit Notes for K11a164's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-7t^{3}+20t^{2}-35t+43-35t^{-1}+20t^{-2}-7t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+z^{6}-2z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 169, 0 }
Jones polynomial	$-q^{5}+5q^{4}-11q^{3}+18q^{2}-24q+28-27q^{-1}+23q^{-2}-17q^{-3}+10q^{-4}-4q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+4z^{6}+a^{4}z^{4}-6a^{2}z^{4}-2z^{4}a^{-2}+5z^{4}+2a^{4}z^{2}-5a^{2}z^{2}+z^{2}+a^{4}-a^{2}+a^{-2}$
Kauffman polynomial (db, data sources)	$3a^{2}z^{10}+3z^{10}+8a^{3}z^{9}+18az^{9}+10z^{9}a^{-1}+8a^{4}z^{8}+16a^{2}z^{8}+14z^{8}a^{-2}+22z^{8}+4a^{5}z^{7}-10a^{3}z^{7}-27az^{7}-2z^{7}a^{-1}+11z^{7}a^{-3}+a^{6}z^{6}-17a^{4}z^{6}-49a^{2}z^{6}-19z^{6}a^{-2}+5z^{6}a^{-4}-55z^{6}-8a^{5}z^{5}-3a^{3}z^{5}+az^{5}-19z^{5}a^{-1}-14z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+13a^{4}z^{4}+41a^{2}z^{4}+6z^{4}a^{-2}-4z^{4}a^{-4}+36z^{4}+5a^{5}z^{3}+5a^{3}z^{3}+8az^{3}+12z^{3}a^{-1}+4z^{3}a^{-3}+a^{6}z^{2}-6a^{4}z^{2}-13a^{2}z^{2}+z^{2}a^{-2}-5z^{2}-a^{5}z-a^{3}z-az-za^{-1}+a^{4}+a^{2}-a^{-2}$
The A2 invariant	$q^{18}-q^{16}+3q^{12}-4q^{10}+4q^{8}-2q^{6}-2q^{4}+4q^{2}-5+6q^{-2}-4q^{-4}+q^{-6}+3q^{-8}-3q^{-10}+3q^{-12}-q^{-14}$
The G2 invariant	$q^{94}-3q^{92}+8q^{90}-16q^{88}+23q^{86}-28q^{84}+20q^{82}+10q^{80}-63q^{78}+139q^{76}-210q^{74}+232q^{72}-166q^{70}-19q^{68}+310q^{66}-612q^{64}+808q^{62}-756q^{60}+371q^{58}+267q^{56}-968q^{54}+1451q^{52}-1462q^{50}+947q^{48}-31q^{46}-947q^{44}+1580q^{42}-1599q^{40}+975q^{38}+18q^{36}-939q^{34}+1370q^{32}-1120q^{30}+316q^{28}+702q^{26}-1454q^{24}+1593q^{22}-1032q^{20}-64q^{18}+1251q^{16}-2080q^{14}+2211q^{12}-1558q^{10}+365q^{8}+973q^{6}-1971q^{4}+2260q^{2}-1767+683q^{-2}+538q^{-4}-1418q^{-6}+1613q^{-8}-1076q^{-10}+123q^{-12}+820q^{-14}-1311q^{-16}+1130q^{-18}-397q^{-20}-574q^{-22}+1335q^{-24}-1563q^{-26}+1201q^{-28}-394q^{-30}-496q^{-32}+1148q^{-34}-1369q^{-36}+1146q^{-38}-627q^{-40}+27q^{-42}+446q^{-44}-687q^{-46}+681q^{-48}-496q^{-50}+252q^{-52}-18q^{-54}-141q^{-56}+203q^{-58}-199q^{-60}+141q^{-62}-73q^{-64}+21q^{-66}+16q^{-68}-28q^{-70}+27q^{-72}-20q^{-74}+10q^{-76}-4q^{-78}+q^{-80}$

Further Quantum Invariants

K11a164 Quantum Invariants.

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["K11a164"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-7t^{3}+20t^{2}-35t+43-35t^{-1}+20t^{-2}-7t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+z^{6}-2z^{4}-2z^{2}+1$

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]=

{ 169, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+5q^{4}-11q^{3}+18q^{2}-24q+28-27q^{-1}+23q^{-2}-17q^{-3}+10q^{-4}-4q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+4z^{6}+a^{4}z^{4}-6a^{2}z^{4}-2z^{4}a^{-2}+5z^{4}+2a^{4}z^{2}-5a^{2}z^{2}+z^{2}+a^{4}-a^{2}+a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{2}z^{10}+3z^{10}+8a^{3}z^{9}+18az^{9}+10z^{9}a^{-1}+8a^{4}z^{8}+16a^{2}z^{8}+14z^{8}a^{-2}+22z^{8}+4a^{5}z^{7}-10a^{3}z^{7}-27az^{7}-2z^{7}a^{-1}+11z^{7}a^{-3}+a^{6}z^{6}-17a^{4}z^{6}-49a^{2}z^{6}-19z^{6}a^{-2}+5z^{6}a^{-4}-55z^{6}-8a^{5}z^{5}-3a^{3}z^{5}+az^{5}-19z^{5}a^{-1}-14z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+13a^{4}z^{4}+41a^{2}z^{4}+6z^{4}a^{-2}-4z^{4}a^{-4}+36z^{4}+5a^{5}z^{3}+5a^{3}z^{3}+8az^{3}+12z^{3}a^{-1}+4z^{3}a^{-3}+a^{6}z^{2}-6a^{4}z^{2}-13a^{2}z^{2}+z^{2}a^{-2}-5z^{2}-a^{5}z-a^{3}z-az-za^{-1}+a^{4}+a^{2}-a^{-2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a164"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]=

{ $t^{4}-7t^{3}+20t^{2}-35t+43-35t^{-1}+20t^{-2}-7t^{-3}+t^{-4}$, $-q^{5}+5q^{4}-11q^{3}+18q^{2}-24q+28-27q^{-1}+23q^{-2}-17q^{-3}+10q^{-4}-4q^{-5}+q^{-6}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-2, 1)

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 $-8$ $8$ $32$ ${\frac {164}{3}}$ ${\frac {100}{3}}$ $-64$ $-{\frac {400}{3}}$ $-{\frac {160}{3}}$ $-24$ $-{\frac {256}{3}}$ $32$ $-{\frac {1312}{3}}$ $-{\frac {800}{3}}$ $-{\frac {4111}{15}}$ ${\frac {908}{5}}$ $-{\frac {21124}{45}}$ ${\frac {655}{9}}$ $-{\frac {1951}{15}}$

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=$0 is the signature of K11a164. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     4   4 7                   7 1   -6 5                 11 4     7 3               13 7       -6 1             15 11         4 -1           13 14           1 -3         10 14             -4 -5       7 13               6 -7     3 10                 -7 -9   1 7                   6 -11   3                     -3 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{15}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a163 K11a165