K11a274

K11a273

K11a275

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a274 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,4,11,3 X14,5,15,6 X18,7,19,8 X2,10,3,9 X20,11,21,12 X22,14,1,13 X4,15,5,16 X12,17,13,18 X8,19,9,20 X16,22,17,21
Gauss code	1, -5, 2, -8, 3, -1, 4, -10, 5, -2, 6, -9, 7, -3, 8, -11, 9, -4, 10, -6, 11, -7
Dowker-Thistlethwaite code	6 10 14 18 2 20 22 4 12 8 16

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:K11a274/ThurstonBennequinNumber Hyperbolic Volume 18.6505 A-Polynomial See Data:K11a274/A-polynomial

[edit Notes for K11a274'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 K11a274's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+18t^{2}-35t+45-35t^{-1}+18t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}+2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 165, 0 }
Jones polynomial	$-q^{5}+4q^{4}-10q^{3}+18q^{2}-23q+27-27q^{-1}+23q^{-2}-17q^{-3}+10q^{-4}-4q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-7a^{2}z^{4}-3z^{4}a^{-2}+11z^{4}+2a^{4}z^{2}-10a^{2}z^{2}-4z^{2}a^{-2}+11z^{2}+2a^{4}-5a^{2}-a^{-2}+5$
Kauffman polynomial (db, data sources)	$3a^{2}z^{10}+3z^{10}+8a^{3}z^{9}+18az^{9}+10z^{9}a^{-1}+8a^{4}z^{8}+16a^{2}z^{8}+13z^{8}a^{-2}+21z^{8}+4a^{5}z^{7}-10a^{3}z^{7}-30az^{7}-7z^{7}a^{-1}+9z^{7}a^{-3}+a^{6}z^{6}-17a^{4}z^{6}-51a^{2}z^{6}-22z^{6}a^{-2}+4z^{6}a^{-4}-59z^{6}-8a^{5}z^{5}-3a^{3}z^{5}+10az^{5}-8z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+13a^{4}z^{4}+48a^{2}z^{4}+19z^{4}a^{-2}-4z^{4}a^{-4}+56z^{4}+5a^{5}z^{3}+4a^{3}z^{3}-az^{3}+8z^{3}a^{-1}+7z^{3}a^{-3}-z^{3}a^{-5}+a^{6}z^{2}-6a^{4}z^{2}-24a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}-27z^{2}-a^{5}z+az-za^{-1}-za^{-3}+2a^{4}+5a^{2}+a^{-2}+5$
The A2 invariant	$q^{18}-q^{16}+q^{14}+3q^{12}-5q^{10}+3q^{8}-3q^{6}-2q^{4}+4q^{2}-4+7q^{-2}-3q^{-4}+2q^{-6}+3q^{-8}-4q^{-10}+2q^{-12}-q^{-14}$
The G2 invariant	Data:K11a274/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a274 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["K11a274"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+18t^{2}-35t+45-35t^{-1}+18t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 165, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-10q^{3}+18q^{2}-23q+27-27q^{-1}+23q^{-2}-17q^{-3}+10q^{-4}-4q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-7a^{2}z^{4}-3z^{4}a^{-2}+11z^{4}+2a^{4}z^{2}-10a^{2}z^{2}-4z^{2}a^{-2}+11z^{2}+2a^{4}-5a^{2}-a^{-2}+5$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{2}z^{10}+3z^{10}+8a^{3}z^{9}+18az^{9}+10z^{9}a^{-1}+8a^{4}z^{8}+16a^{2}z^{8}+13z^{8}a^{-2}+21z^{8}+4a^{5}z^{7}-10a^{3}z^{7}-30az^{7}-7z^{7}a^{-1}+9z^{7}a^{-3}+a^{6}z^{6}-17a^{4}z^{6}-51a^{2}z^{6}-22z^{6}a^{-2}+4z^{6}a^{-4}-59z^{6}-8a^{5}z^{5}-3a^{3}z^{5}+10az^{5}-8z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+13a^{4}z^{4}+48a^{2}z^{4}+19z^{4}a^{-2}-4z^{4}a^{-4}+56z^{4}+5a^{5}z^{3}+4a^{3}z^{3}-az^{3}+8z^{3}a^{-1}+7z^{3}a^{-3}-z^{3}a^{-5}+a^{6}z^{2}-6a^{4}z^{2}-24a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}-27z^{2}-a^{5}z+az-za^{-1}-za^{-3}+2a^{4}+5a^{2}+a^{-2}+5$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11a274"];

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]=

{ $t^{4}-6t^{3}+18t^{2}-35t+45-35t^{-1}+18t^{-2}-6t^{-3}+t^{-4}$, $-q^{5}+4q^{4}-10q^{3}+18q^{2}-23q+27-27q^{-1}+23q^{-2}-17q^{-3}+10q^{-4}-4q^{-5}+q^{-6}$ }

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]=

{}

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, 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 $-4$ $16$ $8$ $-{\frac {62}{3}}$ $-{\frac {34}{3}}$ $-64$ $-{\frac {320}{3}}$ $-{\frac {224}{3}}$ $16$ $-{\frac {32}{3}}$ $128$ ${\frac {248}{3}}$ ${\frac {136}{3}}$ ${\frac {11729}{30}}$ $-{\frac {2258}{15}}$ ${\frac {18658}{45}}$ $-{\frac {689}{18}}$ ${\frac {2609}{30}}$

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 K11a274. 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                   7 1   -6 5                 11 3     8 3               12 7       -5 1             15 11         4 -1           13 13           0 -3         10 14             -4 -5       7 13               6 -7     3 10                 -7 -9   1 7                   6 -11   3                     -3 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{15}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a273 K11a275