K11a163

K11a162

K11a164

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a163 at Knotilus!

Knot presentations

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

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:K11a163/ThurstonBennequinNumber Hyperbolic Volume 15.9193 A-Polynomial See Data:K11a163/A-polynomial

[edit Notes for K11a163's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a163's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-15t^{2}+24t-27+24t^{-1}-15t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}+z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 119, 2 }
Jones polynomial	$q^{7}-4q^{6}+8q^{5}-13q^{4}+17q^{3}-19q^{2}+19q-15+12q^{-1}-7q^{-2}+3q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-9z^{4}a^{-2}+3z^{4}a^{-4}+8z^{4}-3a^{2}z^{2}-7z^{2}a^{-2}+2z^{2}a^{-4}+10z^{2}-2a^{2}-2a^{-2}+5$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+4az^{9}+11z^{9}a^{-1}+7z^{9}a^{-3}+3a^{2}z^{8}+12z^{8}a^{-2}+11z^{8}a^{-4}+4z^{8}+a^{3}z^{7}-11az^{7}-28z^{7}a^{-1}-5z^{7}a^{-3}+11z^{7}a^{-5}-11a^{2}z^{6}-44z^{6}a^{-2}-17z^{6}a^{-4}+8z^{6}a^{-6}-30z^{6}-4a^{3}z^{5}+4az^{5}+13z^{5}a^{-1}-13z^{5}a^{-3}-14z^{5}a^{-5}+4z^{5}a^{-7}+13a^{2}z^{4}+40z^{4}a^{-2}+8z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+37z^{4}+5a^{3}z^{3}+6az^{3}+6z^{3}a^{-1}+12z^{3}a^{-3}+5z^{3}a^{-5}-2z^{3}a^{-7}-7a^{2}z^{2}-14z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}a^{-6}-18z^{2}-2a^{3}z-4az-4za^{-1}-2za^{-3}+2a^{2}+2a^{-2}+5$
The A2 invariant	$-q^{12}-2q^{6}+3q^{4}-q^{2}+3+3q^{-2}-2q^{-4}+4q^{-6}-4q^{-8}+2q^{-10}-q^{-12}-2q^{-14}+2q^{-16}-2q^{-18}+q^{-20}$
The G2 invariant	$q^{60}-2q^{58}+6q^{56}-11q^{54}+15q^{52}-18q^{50}+9q^{48}+12q^{46}-48q^{44}+91q^{42}-123q^{40}+113q^{38}-47q^{36}-78q^{34}+225q^{32}-330q^{30}+336q^{28}-214q^{26}-25q^{24}+288q^{22}-482q^{20}+515q^{18}-346q^{16}+48q^{14}+262q^{12}-454q^{10}+448q^{8}-255q^{6}-33q^{4}+292q^{2}-393+310q^{-2}-59q^{-4}-236q^{-6}+464q^{-8}-507q^{-10}+354q^{-12}-51q^{-14}-304q^{-16}+578q^{-18}-666q^{-20}+535q^{-22}-213q^{-24}-177q^{-26}+493q^{-28}-622q^{-30}+507q^{-32}-224q^{-34}-113q^{-36}+354q^{-38}-407q^{-40}+265q^{-42}-5q^{-44}-237q^{-46}+357q^{-48}-301q^{-50}+97q^{-52}+145q^{-54}-336q^{-56}+400q^{-58}-320q^{-60}+149q^{-62}+55q^{-64}-222q^{-66}+302q^{-68}-296q^{-70}+215q^{-72}-99q^{-74}-16q^{-76}+106q^{-78}-157q^{-80}+161q^{-82}-127q^{-84}+77q^{-86}-19q^{-88}-26q^{-90}+51q^{-92}-61q^{-94}+51q^{-96}-32q^{-98}+15q^{-100}+q^{-102}-8q^{-104}+10q^{-106}-10q^{-108}+6q^{-110}-3q^{-112}+q^{-114}$

Further Quantum Invariants

K11a163 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["K11a163"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-15t^{2}+24t-27+24t^{-1}-15t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 119, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-4q^{6}+8q^{5}-13q^{4}+17q^{3}-19q^{2}+19q-15+12q^{-1}-7q^{-2}+3q^{-3}-q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-9z^{4}a^{-2}+3z^{4}a^{-4}+8z^{4}-3a^{2}z^{2}-7z^{2}a^{-2}+2z^{2}a^{-4}+10z^{2}-2a^{2}-2a^{-2}+5$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+4az^{9}+11z^{9}a^{-1}+7z^{9}a^{-3}+3a^{2}z^{8}+12z^{8}a^{-2}+11z^{8}a^{-4}+4z^{8}+a^{3}z^{7}-11az^{7}-28z^{7}a^{-1}-5z^{7}a^{-3}+11z^{7}a^{-5}-11a^{2}z^{6}-44z^{6}a^{-2}-17z^{6}a^{-4}+8z^{6}a^{-6}-30z^{6}-4a^{3}z^{5}+4az^{5}+13z^{5}a^{-1}-13z^{5}a^{-3}-14z^{5}a^{-5}+4z^{5}a^{-7}+13a^{2}z^{4}+40z^{4}a^{-2}+8z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+37z^{4}+5a^{3}z^{3}+6az^{3}+6z^{3}a^{-1}+12z^{3}a^{-3}+5z^{3}a^{-5}-2z^{3}a^{-7}-7a^{2}z^{2}-14z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}a^{-6}-18z^{2}-2a^{3}z-4az-4za^{-1}-2za^{-3}+2a^{2}+2a^{-2}+5$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a66,}

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["K11a163"];

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]=

{ $-t^{4}+6t^{3}-15t^{2}+24t-27+24t^{-1}-15t^{-2}+6t^{-3}-t^{-4}$, $q^{7}-4q^{6}+8q^{5}-13q^{4}+17q^{3}-19q^{2}+19q-15+12q^{-1}-7q^{-2}+3q^{-3}-q^{-4}$ }

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]=

{K11a66,}

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, 0)

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 $8$ $0$ $32$ ${\frac {124}{3}}$ ${\frac {20}{3}}$ $0$ $0$ $64$ $-64$ ${\frac {256}{3}}$ $0$ ${\frac {992}{3}}$ ${\frac {160}{3}}$ ${\frac {5071}{15}}$ ${\frac {2516}{15}}$ $-{\frac {2516}{45}}$ ${\frac {305}{9}}$ $-{\frac {209}{15}}$

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 K11a163. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   5 1   4 9                 8 3     -5 7               9 5       4 5             10 8         -2 3           9 9           0 1         7 11             4 -1       5 8               -3 -3     2 7                 5 -5   1 5                   -4 -7   2                     2 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a162 K11a164