K11a259

K11a258

K11a260

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a259 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a259/ThurstonBennequinNumber Hyperbolic Volume 12.5201 A-Polynomial See Data:K11a259/A-polynomial

[edit Notes for K11a259's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -6

[edit Notes for K11a259's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-10t^{2}+15t-17+15t^{-1}-10t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 79, 6 }
Jones polynomial	$-q^{12}+3q^{11}-6q^{10}+9q^{9}-11q^{8}+12q^{7}-12q^{6}+10q^{5}-7q^{4}+5q^{3}-2q^{2}+q$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-6}+z^{6}a^{-4}-6z^{6}a^{-6}+2z^{6}a^{-8}+5z^{4}a^{-4}-13z^{4}a^{-6}+9z^{4}a^{-8}-z^{4}a^{-10}+8z^{2}a^{-4}-13z^{2}a^{-6}+12z^{2}a^{-8}-3z^{2}a^{-10}+4a^{-4}-6a^{-6}+5a^{-8}-2a^{-10}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-6}+z^{10}a^{-8}+2z^{9}a^{-5}+6z^{9}a^{-7}+4z^{9}a^{-9}+z^{8}a^{-4}+z^{8}a^{-6}+8z^{8}a^{-8}+8z^{8}a^{-10}-10z^{7}a^{-5}-22z^{7}a^{-7}-2z^{7}a^{-9}+10z^{7}a^{-11}-6z^{6}a^{-4}-22z^{6}a^{-6}-42z^{6}a^{-8}-17z^{6}a^{-10}+9z^{6}a^{-12}+15z^{5}a^{-5}+17z^{5}a^{-7}-24z^{5}a^{-9}-20z^{5}a^{-11}+6z^{5}a^{-13}+13z^{4}a^{-4}+44z^{4}a^{-6}+53z^{4}a^{-8}+6z^{4}a^{-10}-13z^{4}a^{-12}+3z^{4}a^{-14}-5z^{3}a^{-5}+6z^{3}a^{-7}+26z^{3}a^{-9}+10z^{3}a^{-11}-4z^{3}a^{-13}+z^{3}a^{-15}-12z^{2}a^{-4}-28z^{2}a^{-6}-25z^{2}a^{-8}-4z^{2}a^{-10}+5z^{2}a^{-12}-2za^{-5}-5za^{-7}-7za^{-9}-3za^{-11}+za^{-13}+4a^{-4}+6a^{-6}+5a^{-8}+2a^{-10}$
The A2 invariant	$q^{-4}+2q^{-8}+q^{-10}+2q^{-14}-3q^{-16}+q^{-18}-2q^{-20}+2q^{-24}-q^{-26}+2q^{-28}-q^{-30}-q^{-36}$
The G2 invariant	Data:K11a259/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a259 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["K11a259"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-10t^{2}+15t-17+15t^{-1}-10t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-3z^{6}+4z^{2}+1$

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]=

{ 79, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{12}+3q^{11}-6q^{10}+9q^{9}-11q^{8}+12q^{7}-12q^{6}+10q^{5}-7q^{4}+5q^{3}-2q^{2}+q$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-6}+z^{6}a^{-4}-6z^{6}a^{-6}+2z^{6}a^{-8}+5z^{4}a^{-4}-13z^{4}a^{-6}+9z^{4}a^{-8}-z^{4}a^{-10}+8z^{2}a^{-4}-13z^{2}a^{-6}+12z^{2}a^{-8}-3z^{2}a^{-10}+4a^{-4}-6a^{-6}+5a^{-8}-2a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-6}+z^{10}a^{-8}+2z^{9}a^{-5}+6z^{9}a^{-7}+4z^{9}a^{-9}+z^{8}a^{-4}+z^{8}a^{-6}+8z^{8}a^{-8}+8z^{8}a^{-10}-10z^{7}a^{-5}-22z^{7}a^{-7}-2z^{7}a^{-9}+10z^{7}a^{-11}-6z^{6}a^{-4}-22z^{6}a^{-6}-42z^{6}a^{-8}-17z^{6}a^{-10}+9z^{6}a^{-12}+15z^{5}a^{-5}+17z^{5}a^{-7}-24z^{5}a^{-9}-20z^{5}a^{-11}+6z^{5}a^{-13}+13z^{4}a^{-4}+44z^{4}a^{-6}+53z^{4}a^{-8}+6z^{4}a^{-10}-13z^{4}a^{-12}+3z^{4}a^{-14}-5z^{3}a^{-5}+6z^{3}a^{-7}+26z^{3}a^{-9}+10z^{3}a^{-11}-4z^{3}a^{-13}+z^{3}a^{-15}-12z^{2}a^{-4}-28z^{2}a^{-6}-25z^{2}a^{-8}-4z^{2}a^{-10}+5z^{2}a^{-12}-2za^{-5}-5za^{-7}-7za^{-9}-3za^{-11}+za^{-13}+4a^{-4}+6a^{-6}+5a^{-8}+2a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a221,}

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["K11a259"];

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]=

{ $-t^{4}+5t^{3}-10t^{2}+15t-17+15t^{-1}-10t^{-2}+5t^{-3}-t^{-4}$, $-q^{12}+3q^{11}-6q^{10}+9q^{9}-11q^{8}+12q^{7}-12q^{6}+10q^{5}-7q^{4}+5q^{3}-2q^{2}+q$ }

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]=

{K11a221,}

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:

(4, 10)

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 $16$ $80$ $128$ ${\frac {1352}{3}}$ ${\frac {184}{3}}$ $1280$ ${\frac {8192}{3}}$ ${\frac {1184}{3}}$ $400$ ${\frac {2048}{3}}$ $3200$ ${\frac {21632}{3}}$ ${\frac {2944}{3}}$ ${\frac {256262}{15}}$ $-{\frac {1768}{15}}$ ${\frac {307448}{45}}$ ${\frac {2698}{9}}$ ${\frac {12662}{15}}$

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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=$6 is the signature of K11a259. 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 8 9 χ 25                       1 -1 23                     2   2 21                   4 1   -3 19                 5 2     3 17               6 4       -2 15             6 5         1 13           6 6           0 11         4 6             -2 9       3 6               3 7     2 4                 -2 5   1 4                   3 3   1                     -1 1 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a258 K11a260