K11a252

K11a251

K11a253

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a252 at Knotilus!

Knot presentations

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

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:K11a252/ThurstonBennequinNumber Hyperbolic Volume 16.8559 A-Polynomial See Data:K11a252/A-polynomial

[edit Notes for K11a252'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 K11a252's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-16t^{2}+27t-31+27t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 131, 2 }
Jones polynomial	$q^{7}-4q^{6}+9q^{5}-15q^{4}+19q^{3}-21q^{2}+21q-17+13q^{-1}-7q^{-2}+3q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-10z^{4}a^{-2}+3z^{4}a^{-4}+8z^{4}-3a^{2}z^{2}-10z^{2}a^{-2}+3z^{2}a^{-4}+11z^{2}-2a^{2}-4a^{-2}+a^{-4}+6$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+4az^{9}+12z^{9}a^{-1}+8z^{9}a^{-3}+3a^{2}z^{8}+17z^{8}a^{-2}+14z^{8}a^{-4}+6z^{8}+a^{3}z^{7}-10az^{7}-26z^{7}a^{-1}-z^{7}a^{-3}+14z^{7}a^{-5}-11a^{2}z^{6}-55z^{6}a^{-2}-22z^{6}a^{-4}+9z^{6}a^{-6}-35z^{6}-4a^{3}z^{5}+2az^{5}+3z^{5}a^{-1}-27z^{5}a^{-3}-20z^{5}a^{-5}+4z^{5}a^{-7}+14a^{2}z^{4}+47z^{4}a^{-2}+10z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+43z^{4}+5a^{3}z^{3}+9az^{3}+15z^{3}a^{-1}+23z^{3}a^{-3}+11z^{3}a^{-5}-z^{3}a^{-7}-8a^{2}z^{2}-19z^{2}a^{-2}-3z^{2}a^{-4}+2z^{2}a^{-6}-22z^{2}-2a^{3}z-5az-7za^{-1}-6za^{-3}-2za^{-5}+2a^{2}+4a^{-2}+a^{-4}+6$
The A2 invariant	$-q^{12}-2q^{6}+4q^{4}-q^{2}+3+3q^{-2}-3q^{-4}+4q^{-6}-5q^{-8}+2q^{-10}-q^{-12}-2q^{-14}+3q^{-16}-2q^{-18}+q^{-20}$
The G2 invariant	Data:K11a252/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a252 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["K11a252"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-16t^{2}+27t-31+27t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 131, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-4q^{6}+9q^{5}-15q^{4}+19q^{3}-21q^{2}+21q-17+13q^{-1}-7q^{-2}+3q^{-3}-q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-10z^{4}a^{-2}+3z^{4}a^{-4}+8z^{4}-3a^{2}z^{2}-10z^{2}a^{-2}+3z^{2}a^{-4}+11z^{2}-2a^{2}-4a^{-2}+a^{-4}+6$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+4az^{9}+12z^{9}a^{-1}+8z^{9}a^{-3}+3a^{2}z^{8}+17z^{8}a^{-2}+14z^{8}a^{-4}+6z^{8}+a^{3}z^{7}-10az^{7}-26z^{7}a^{-1}-z^{7}a^{-3}+14z^{7}a^{-5}-11a^{2}z^{6}-55z^{6}a^{-2}-22z^{6}a^{-4}+9z^{6}a^{-6}-35z^{6}-4a^{3}z^{5}+2az^{5}+3z^{5}a^{-1}-27z^{5}a^{-3}-20z^{5}a^{-5}+4z^{5}a^{-7}+14a^{2}z^{4}+47z^{4}a^{-2}+10z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+43z^{4}+5a^{3}z^{3}+9az^{3}+15z^{3}a^{-1}+23z^{3}a^{-3}+11z^{3}a^{-5}-z^{3}a^{-7}-8a^{2}z^{2}-19z^{2}a^{-2}-3z^{2}a^{-4}+2z^{2}a^{-6}-22z^{2}-2a^{3}z-5az-7za^{-1}-6za^{-3}-2za^{-5}+2a^{2}+4a^{-2}+a^{-4}+6$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a131, K11a254,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a254,}

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["K11a252"];

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]=

{ $-t^{4}+6t^{3}-16t^{2}+27t-31+27t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$, $q^{7}-4q^{6}+9q^{5}-15q^{4}+19q^{3}-21q^{2}+21q-17+13q^{-1}-7q^{-2}+3q^{-3}-q^{-4}$ }

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]=

{K11a131, K11a254,}

In[6]:=

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

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

Out[6]=

{K11a254,}

Vassiliev invariants

V2 and V3:

(1, -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 $4$ $-8$ $8$ ${\frac {14}{3}}$ ${\frac {10}{3}}$ $-32$ $-{\frac {80}{3}}$ ${\frac {160}{3}}$ $-40$ ${\frac {32}{3}}$ $32$ ${\frac {56}{3}}$ ${\frac {40}{3}}$ ${\frac {2911}{30}}$ ${\frac {1658}{15}}$ $-{\frac {3418}{45}}$ ${\frac {737}{18}}$ $-{\frac {449}{30}}$

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 K11a252. 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                   6 1   5 9                 9 3     -6 7               10 6       4 5             11 9         -2 3           10 10           0 1         8 12             4 -1       5 9               -4 -3     2 8                 6 -5   1 5                   -4 -7   2                     2 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a251 K11a253