K11n12

K11n11

K11n13

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n12 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X7,14,8,15 X2,9,3,10 X11,19,12,18 X13,6,14,7 X15,1,16,22 X17,21,18,20 X19,13,20,12 X21,17,22,16
Gauss code	1, -5, 2, -1, 3, 7, -4, -2, 5, -3, -6, 10, -7, 4, -8, 11, -9, 6, -10, 9, -11, 8
Dowker-Thistlethwaite code	4 8 10 -14 2 -18 -6 -22 -20 -12 -16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n12/ThurstonBennequinNumber Hyperbolic Volume 8.07031 A-Polynomial See Data:K11n12/A-polynomial

[edit Notes for K11n12's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n12's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{2}-3t+5-3t^{-1}+t^{-2}$
Conway polynomial	$z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 13, 0 }
Jones polynomial	$-q^{7}+2q^{6}-2q^{5}+2q^{4}-2q^{3}+2q^{2}-q+1$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}-z^{2}a^{-2}+3z^{2}a^{-4}-z^{2}a^{-6}-a^{-2}+2a^{-4}-a^{-6}+1$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}a^{-6}-5z^{7}a^{-3}-4z^{7}a^{-5}+z^{7}a^{-7}-6z^{6}a^{-2}-17z^{6}a^{-4}-11z^{6}a^{-6}+5z^{5}a^{-3}-5z^{5}a^{-7}+10z^{4}a^{-2}+26z^{4}a^{-4}+16z^{4}a^{-6}-z^{3}a^{-1}+7z^{3}a^{-5}+6z^{3}a^{-7}-7z^{2}a^{-2}-13z^{2}a^{-4}-7z^{2}a^{-6}-z^{2}-za^{-3}-3za^{-5}-2za^{-7}+a^{-2}+2a^{-4}+a^{-6}+1$
The A2 invariant	Data:K11n12/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n12/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n12 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["K11n12"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{2}-3t+5-3t^{-1}+t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{4}+z^{2}+1$

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]=

{ 13, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{7}+2q^{6}-2q^{5}+2q^{4}-2q^{3}+2q^{2}-q+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}a^{-6}-5z^{7}a^{-3}-4z^{7}a^{-5}+z^{7}a^{-7}-6z^{6}a^{-2}-17z^{6}a^{-4}-11z^{6}a^{-6}+5z^{5}a^{-3}-5z^{5}a^{-7}+10z^{4}a^{-2}+26z^{4}a^{-4}+16z^{4}a^{-6}-z^{3}a^{-1}+7z^{3}a^{-5}+6z^{3}a^{-7}-7z^{2}a^{-2}-13z^{2}a^{-4}-7z^{2}a^{-6}-z^{2}-za^{-3}-3za^{-5}-2za^{-7}+a^{-2}+2a^{-4}+a^{-6}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {6_3,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {9_43,}

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["K11n12"];

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]=

{ $t^{2}-3t+5-3t^{-1}+t^{-2}$, $-q^{7}+2q^{6}-2q^{5}+2q^{4}-2q^{3}+2q^{2}-q+1$ }

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]=

{6_3,}

In[6]:=

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

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

Out[6]=

{9_43,}

Vassiliev invariants

V2 and V3:

(1, 2)

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 $4$ $16$ $8$ ${\frac {110}{3}}$ $-{\frac {14}{3}}$ $64$ ${\frac {352}{3}}$ 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 -\frac{32}{3}} $16$ ${\frac {32}{3}}$ $128$ ${\frac {440}{3}}$ $-{\frac {56}{3}}$ ${\frac {13471}{30}}$ ${\frac {338}{15}}$ ${\frac {3902}{45}}$ ${\frac {65}{18}}$ $-{\frac {449}{30}}$

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=$0 is the signature of K11n12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n11 K11n13