K11a263

K11a262

K11a264

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a263 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X8493 X16,6,17,5 X2837 X20,10,21,9 X22,12,1,11 X18,14,19,13 X4,16,5,15 X12,18,13,17 X14,20,15,19 X10,22,11,21
Gauss code	1, -4, 2, -8, 3, -1, 4, -2, 5, -11, 6, -9, 7, -10, 8, -3, 9, -7, 10, -5, 11, -6
Dowker-Thistlethwaite code	6 8 16 2 20 22 18 4 12 14 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 4 3-genus 4 Bridge index 4 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a263/ThurstonBennequinNumber Hyperbolic Volume 13.7204 A-Polynomial See Data:K11a263/A-polynomial

[edit Notes for K11a263's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -8

[edit Notes for K11a263's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{4}-6t^{3}+11t^{2}-14t+15-14t^{-1}+11t^{-2}-6t^{-3}+2t^{-4}$
Conway polynomial	$2z^{8}+10z^{6}+15z^{4}+8z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{2}-t+1\right\}$
Determinant and Signature	{ 81, 8 }
Jones polynomial	$-q^{15}+3q^{14}-6q^{13}+10q^{12}-12q^{11}+12q^{10}-13q^{9}+10q^{8}-7q^{7}+5q^{6}-q^{5}+q^{4}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+7z^{6}a^{-8}+4z^{6}a^{-10}-z^{6}a^{-12}+18z^{4}a^{-8}-3z^{4}a^{-12}+20z^{2}a^{-8}-13z^{2}a^{-10}+z^{2}a^{-12}+8a^{-8}-11a^{-10}+5a^{-12}-a^{-14}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-10}+z^{10}a^{-12}+z^{9}a^{-9}+5z^{9}a^{-11}+4z^{9}a^{-13}+z^{8}a^{-8}-z^{8}a^{-10}+7z^{8}a^{-12}+9z^{8}a^{-14}-3z^{7}a^{-9}-14z^{7}a^{-11}+z^{7}a^{-13}+12z^{7}a^{-15}-7z^{6}a^{-8}-11z^{6}a^{-10}-32z^{6}a^{-12}-18z^{6}a^{-14}+10z^{6}a^{-16}-3z^{5}a^{-9}-6z^{5}a^{-11}-36z^{5}a^{-13}-27z^{5}a^{-15}+6z^{5}a^{-17}+18z^{4}a^{-8}+28z^{4}a^{-10}+28z^{4}a^{-12}-15z^{4}a^{-16}+3z^{4}a^{-18}+16z^{3}a^{-9}+37z^{3}a^{-11}+43z^{3}a^{-13}+18z^{3}a^{-15}-3z^{3}a^{-17}+z^{3}a^{-19}-20z^{2}a^{-8}-26z^{2}a^{-10}-7z^{2}a^{-12}+5z^{2}a^{-14}+6z^{2}a^{-16}-12za^{-9}-22za^{-11}-16za^{-13}-6za^{-15}+8a^{-8}+11a^{-10}+5a^{-12}+a^{-14}$
The A2 invariant	$q^{-14}+4q^{-18}+q^{-20}+4q^{-22}+q^{-24}-2q^{-26}-q^{-28}-7q^{-30}-2q^{-34}+2q^{-36}+3q^{-38}+q^{-42}-2q^{-44}$
The G2 invariant	Data:K11a263/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a263 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["K11a263"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{4}-6t^{3}+11t^{2}-14t+15-14t^{-1}+11t^{-2}-6t^{-3}+2t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{8}+10z^{6}+15z^{4}+8z^{2}+1$

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]=

$\left\{t^{2}-t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 81, 8 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{15}+3q^{14}-6q^{13}+10q^{12}-12q^{11}+12q^{10}-13q^{9}+10q^{8}-7q^{7}+5q^{6}-q^{5}+q^{4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-8}+z^{8}a^{-10}+7z^{6}a^{-8}+4z^{6}a^{-10}-z^{6}a^{-12}+18z^{4}a^{-8}-3z^{4}a^{-12}+20z^{2}a^{-8}-13z^{2}a^{-10}+z^{2}a^{-12}+8a^{-8}-11a^{-10}+5a^{-12}-a^{-14}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-10}+z^{10}a^{-12}+z^{9}a^{-9}+5z^{9}a^{-11}+4z^{9}a^{-13}+z^{8}a^{-8}-z^{8}a^{-10}+7z^{8}a^{-12}+9z^{8}a^{-14}-3z^{7}a^{-9}-14z^{7}a^{-11}+z^{7}a^{-13}+12z^{7}a^{-15}-7z^{6}a^{-8}-11z^{6}a^{-10}-32z^{6}a^{-12}-18z^{6}a^{-14}+10z^{6}a^{-16}-3z^{5}a^{-9}-6z^{5}a^{-11}-36z^{5}a^{-13}-27z^{5}a^{-15}+6z^{5}a^{-17}+18z^{4}a^{-8}+28z^{4}a^{-10}+28z^{4}a^{-12}-15z^{4}a^{-16}+3z^{4}a^{-18}+16z^{3}a^{-9}+37z^{3}a^{-11}+43z^{3}a^{-13}+18z^{3}a^{-15}-3z^{3}a^{-17}+z^{3}a^{-19}-20z^{2}a^{-8}-26z^{2}a^{-10}-7z^{2}a^{-12}+5z^{2}a^{-14}+6z^{2}a^{-16}-12za^{-9}-22za^{-11}-16za^{-13}-6za^{-15}+8a^{-8}+11a^{-10}+5a^{-12}+a^{-14}$

"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["K11a263"];

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]=

{ $2t^{4}-6t^{3}+11t^{2}-14t+15-14t^{-1}+11t^{-2}-6t^{-3}+2t^{-4}$, $-q^{15}+3q^{14}-6q^{13}+10q^{12}-12q^{11}+12q^{10}-13q^{9}+10q^{8}-7q^{7}+5q^{6}-q^{5}+q^{4}$ }

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]=

{}

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:

(8, 21)

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 $32$ $168$ $512$ ${\frac {3424}{3}}$ ${\frac {392}{3}}$ $5376$ $8720$ $1344$ $904$ ${\frac {16384}{3}}$ $14112$ ${\frac {109568}{3}}$ ${\frac {12544}{3}}$ ${\frac {1035844}{15}}$ ${\frac {56744}{15}}$ ${\frac {969736}{45}}$ ${\frac {5804}{9}}$ ${\frac {37444}{15}}$

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=$8 is the signature of K11a263. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 χ 31                       1 -1 29                     2   2 27                   4 1   -3 25                 6 2     4 23               6 4       -2 21             6 6         0 19           7 6           -1 17         3 6             -3 15       4 7               3 13     1 3                 -2 11     4                   4 9 1 1                     0 7 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a262 K11a264