K11n51

K11n50

K11n52

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n51 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,6,15,5 X2837 X9,16,10,17 X11,19,12,18 X6,14,7,13 X15,22,16,1 X17,20,18,21 X19,11,20,10 X21,12,22,13
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, -5, 10, -6, 11, 7, -3, -8, 5, -9, 6, -10, 9, -11, 8
Dowker-Thistlethwaite code	4 8 14 2 -16 -18 6 -22 -20 -10 -12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n51/ThurstonBennequinNumber Hyperbolic Volume 9.50422 A-Polynomial See Data:K11n51/A-polynomial

[edit Notes for K11n51's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11n51's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+4t^{2}-6t+7-6t^{-1}+4t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 29, 0 }
Jones polynomial	$-q^{7}+2q^{6}-3q^{5}+4q^{4}-4q^{3}+5q^{2}-4q+3-2q^{-1}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-5z^{4}a^{-2}+2z^{4}a^{-4}+z^{4}-8z^{2}a^{-2}+7z^{2}a^{-4}-z^{2}a^{-6}+3z^{2}-4a^{-2}+5a^{-4}-2a^{-6}+2$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+4z^{8}a^{-4}+2z^{8}a^{-6}+z^{7}a^{-1}-2z^{7}a^{-3}-2z^{7}a^{-5}+z^{7}a^{-7}-9z^{6}a^{-2}-19z^{6}a^{-4}-10z^{6}a^{-6}-2z^{5}a^{-1}-4z^{5}a^{-3}-7z^{5}a^{-5}-5z^{5}a^{-7}+16z^{4}a^{-2}+27z^{4}a^{-4}+14z^{4}a^{-6}+3z^{4}+2az^{3}+2z^{3}a^{-1}+7z^{3}a^{-3}+14z^{3}a^{-5}+7z^{3}a^{-7}+a^{2}z^{2}-15z^{2}a^{-2}-17z^{2}a^{-4}-7z^{2}a^{-6}-4z^{2}-az-2za^{-1}-3za^{-3}-5za^{-5}-3za^{-7}+4a^{-2}+5a^{-4}+2a^{-6}+2$
The A2 invariant	Data:K11n51/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n51/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n51 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["K11n51"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 29, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+4z^{8}a^{-4}+2z^{8}a^{-6}+z^{7}a^{-1}-2z^{7}a^{-3}-2z^{7}a^{-5}+z^{7}a^{-7}-9z^{6}a^{-2}-19z^{6}a^{-4}-10z^{6}a^{-6}-2z^{5}a^{-1}-4z^{5}a^{-3}-7z^{5}a^{-5}-5z^{5}a^{-7}+16z^{4}a^{-2}+27z^{4}a^{-4}+14z^{4}a^{-6}+3z^{4}+2az^{3}+2z^{3}a^{-1}+7z^{3}a^{-3}+14z^{3}a^{-5}+7z^{3}a^{-7}+a^{2}z^{2}-15z^{2}a^{-2}-17z^{2}a^{-4}-7z^{2}a^{-6}-4z^{2}-az-2za^{-1}-3za^{-3}-5za^{-5}-3za^{-7}+4a^{-2}+5a^{-4}+2a^{-6}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_127, 10_150,}

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["K11n51"];

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]=

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

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]=

{10_127, 10_150,}

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:

(1, 3)

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 $4$ $24$ $8$ ${\frac {254}{3}}$ ${\frac {58}{3}}$ $96$ $272$ $32$ $56$ ${\frac {32}{3}}$ $288$ ${\frac {1016}{3}}$ ${\frac {232}{3}}$ ${\frac {34831}{30}}$ $-{\frac {742}{15}}$ ${\frac {24422}{45}}$ ${\frac {497}{18}}$ ${\frac {1711}{30}}$

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 K11n51. 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 χ 15                   1 -1 13                 1   1 11               2 1   -1 9             2 1     1 7           2 2       0 5         3 2         1 3       1 2           1 1     2 3             -1 -1   1 2               1 -3   1                 -1 -5 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n50 K11n52