K11a251

K11a250

K11a252

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a251 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X8493 X12,5,13,6 X2837 X16,10,17,9 X18,11,19,12 X4,13,5,14 X20,15,21,16 X22,18,1,17 X14,19,15,20 X10,21,11,22
Gauss code	1, -4, 2, -7, 3, -1, 4, -2, 5, -11, 6, -3, 7, -10, 8, -5, 9, -6, 10, -8, 11, -9
Dowker-Thistlethwaite code	6 8 12 2 16 18 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:K11a251/ThurstonBennequinNumber Hyperbolic Volume 16.9235 A-Polynomial See Data:K11a251/A-polynomial

[edit Notes for K11a251's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a251's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+16t^{2}-27t+33-27t^{-1}+16t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 133, 0 }
Jones polynomial	$-q^{5}+4q^{4}-9q^{3}+15q^{2}-19q+22-21q^{-1}+18q^{-2}-13q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-8a^{2}z^{4}-3z^{4}a^{-2}+10z^{4}+3a^{4}z^{2}-11a^{2}z^{2}-3z^{2}a^{-2}+10z^{2}+2a^{4}-5a^{2}-a^{-2}+5$
Kauffman polynomial (db, data sources)	$2a^{2}z^{10}+2z^{10}+5a^{3}z^{9}+12az^{9}+7z^{9}a^{-1}+5a^{4}z^{8}+9a^{2}z^{8}+10z^{8}a^{-2}+14z^{8}+3a^{5}z^{7}-8a^{3}z^{7}-24az^{7}-5z^{7}a^{-1}+8z^{7}a^{-3}+a^{6}z^{6}-12a^{4}z^{6}-34a^{2}z^{6}-17z^{6}a^{-2}+4z^{6}a^{-4}-42z^{6}-8a^{5}z^{5}+a^{3}z^{5}+16az^{5}-6z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+10a^{4}z^{4}+42a^{2}z^{4}+12z^{4}a^{-2}-5z^{4}a^{-4}+46z^{4}+6a^{5}z^{3}+4a^{3}z^{3}-az^{3}+7z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+2a^{6}z^{2}-6a^{4}z^{2}-24a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}-23z^{2}-2a^{5}z-3a^{3}z-2az-2za^{-1}-za^{-3}+2a^{4}+5a^{2}+a^{-2}+5$
The A2 invariant	$q^{18}+2q^{12}-4q^{10}+2q^{8}-2q^{6}-2q^{4}+4q^{2}-3+6q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-3q^{-10}+2q^{-12}-q^{-14}$
The G2 invariant	Data:K11a251/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a251 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["K11a251"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+16t^{2}-27t+33-27t^{-1}+16t^{-2}-6t^{-3}+t^{-4}$

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]=

{ 133, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-9q^{3}+15q^{2}-19q+22-21q^{-1}+18q^{-2}-13q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{2}z^{10}+2z^{10}+5a^{3}z^{9}+12az^{9}+7z^{9}a^{-1}+5a^{4}z^{8}+9a^{2}z^{8}+10z^{8}a^{-2}+14z^{8}+3a^{5}z^{7}-8a^{3}z^{7}-24az^{7}-5z^{7}a^{-1}+8z^{7}a^{-3}+a^{6}z^{6}-12a^{4}z^{6}-34a^{2}z^{6}-17z^{6}a^{-2}+4z^{6}a^{-4}-42z^{6}-8a^{5}z^{5}+a^{3}z^{5}+16az^{5}-6z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+10a^{4}z^{4}+42a^{2}z^{4}+12z^{4}a^{-2}-5z^{4}a^{-4}+46z^{4}+6a^{5}z^{3}+4a^{3}z^{3}-az^{3}+7z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+2a^{6}z^{2}-6a^{4}z^{2}-24a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}-23z^{2}-2a^{5}z-3a^{3}z-2az-2za^{-1}-za^{-3}+2a^{4}+5a^{2}+a^{-2}+5$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a253,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a228, K11a253,}

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["K11a251"];

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]=

{ $t^{4}-6t^{3}+16t^{2}-27t+33-27t^{-1}+16t^{-2}-6t^{-3}+t^{-4}$, $-q^{5}+4q^{4}-9q^{3}+15q^{2}-19q+22-21q^{-1}+18q^{-2}-13q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$ }

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]=

{K11a253,}

In[6]:=

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

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

Out[6]=

{K11a228, K11a253,}

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 $-4$ $16$ $8$ $-{\frac {14}{3}}$ ${\frac {14}{3}}$ $-64$ $-{\frac {416}{3}}$ $-{\frac {224}{3}}$ $-16$ $-{\frac {32}{3}}$ $128$ ${\frac {56}{3}}$ $-{\frac {56}{3}}$ ${\frac {11009}{30}}$ $-{\frac {1378}{15}}$ ${\frac {13378}{45}}$ $-{\frac {65}{18}}$ ${\frac {1409}{30}}$

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 K11a251. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   6 1   -5 5                 9 3     6 3               10 6       -4 1             12 9         3 -1           10 11           1 -3         8 11             -3 -5       5 10               5 -7     2 8                 -6 -9   1 5                   4 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a250 K11a252