K11a156

K11a155

K11a157

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a156 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X18,5,19,6 X12,8,13,7 X14,10,15,9 X2,11,3,12 X8,14,9,13 X20,15,21,16 X22,17,1,18 X6,19,7,20 X16,21,17,22
Gauss code	1, -6, 2, -1, 3, -10, 4, -7, 5, -2, 6, -4, 7, -5, 8, -11, 9, -3, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 10 18 12 14 2 8 20 22 6 16

A Braid Representative	{{{braid_table}}}
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:K11a156/ThurstonBennequinNumber Hyperbolic Volume 14.0856 A-Polynomial See Data:K11a156/A-polynomial

[edit Notes for K11a156'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 K11a156's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-11t^{2}+18t-21+18t^{-1}-11t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 91, -2 }
Jones polynomial	$q^{3}-3q^{2}+6q-10+13q^{-1}-14q^{-2}+15q^{-3}-12q^{-4}+9q^{-5}-5q^{-6}+2q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-6a^{2}z^{6}+z^{6}-a^{6}z^{4}+10a^{4}z^{4}-14a^{2}z^{4}+4z^{4}-4a^{6}z^{2}+17a^{4}z^{2}-15a^{2}z^{2}+5z^{2}-4a^{6}+9a^{4}-6a^{2}+2$
Kauffman polynomial (db, data sources)	$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+6a^{3}z^{9}+3az^{9}+4a^{6}z^{8}+6a^{4}z^{8}+6a^{2}z^{8}+4z^{8}+3a^{7}z^{7}-4a^{5}z^{7}-13a^{3}z^{7}-3az^{7}+3z^{7}a^{-1}+2a^{8}z^{6}-10a^{6}z^{6}-25a^{4}z^{6}-24a^{2}z^{6}+z^{6}a^{-2}-10z^{6}+a^{9}z^{5}-5a^{7}z^{5}+7a^{3}z^{5}-8az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+16a^{6}z^{4}+43a^{4}z^{4}+32a^{2}z^{4}-3z^{4}a^{-2}+6z^{4}-3a^{9}z^{3}+2a^{7}z^{3}+10a^{5}z^{3}+8a^{3}z^{3}+10az^{3}+7z^{3}a^{-1}+a^{8}z^{2}-13a^{6}z^{2}-30a^{4}z^{2}-22a^{2}z^{2}+2z^{2}a^{-2}-4z^{2}+2a^{9}z-a^{7}z-6a^{5}z-6a^{3}z-5az-2za^{-1}+4a^{6}+9a^{4}+6a^{2}+2$
The A2 invariant	$-q^{24}-q^{22}-q^{20}-2q^{18}+3q^{16}+3q^{12}+3q^{10}-q^{8}+3q^{6}-4q^{4}+q^{2}-1-q^{-2}+2q^{-4}-q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-q^{126}+3q^{124}-4q^{122}+4q^{120}-3q^{118}-q^{116}+7q^{114}-13q^{112}+19q^{110}-22q^{108}+18q^{106}-9q^{104}-9q^{102}+32q^{100}-54q^{98}+66q^{96}-69q^{94}+44q^{92}-5q^{90}-54q^{88}+112q^{86}-150q^{84}+147q^{82}-100q^{80}+4q^{78}+100q^{76}-183q^{74}+210q^{72}-158q^{70}+50q^{68}+73q^{66}-160q^{64}+172q^{62}-100q^{60}-16q^{58}+133q^{56}-179q^{54}+140q^{52}-14q^{50}-128q^{48}+248q^{46}-272q^{44}+200q^{42}-47q^{40}-126q^{38}+268q^{36}-318q^{34}+269q^{32}-130q^{30}-44q^{28}+185q^{26}-253q^{24}+216q^{22}-105q^{20}-41q^{18}+145q^{16}-177q^{14}+114q^{12}+8q^{10}-135q^{8}+203q^{6}-185q^{4}+81q^{2}+55-173q^{-2}+231q^{-4}-201q^{-6}+115q^{-8}-100q^{-12}+152q^{-14}-149q^{-16}+107q^{-18}-44q^{-20}-11q^{-22}+47q^{-24}-60q^{-26}+53q^{-28}-33q^{-30}+15q^{-32}+q^{-34}-10q^{-36}+10q^{-38}-9q^{-40}+5q^{-42}-2q^{-44}+q^{-46}$

Further Quantum Invariants

K11a156 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["K11a156"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-11t^{2}+18t-21+18t^{-1}-11t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-3z^{6}-z^{4}+3z^{2}+1$

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]=

{ 91, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-3q^{2}+6q-10+13q^{-1}-14q^{-2}+15q^{-3}-12q^{-4}+9q^{-5}-5q^{-6}+2q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+2a^{4}z^{6}-6a^{2}z^{6}+z^{6}-a^{6}z^{4}+10a^{4}z^{4}-14a^{2}z^{4}+4z^{4}-4a^{6}z^{2}+17a^{4}z^{2}-15a^{2}z^{2}+5z^{2}-4a^{6}+9a^{4}-6a^{2}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+6a^{3}z^{9}+3az^{9}+4a^{6}z^{8}+6a^{4}z^{8}+6a^{2}z^{8}+4z^{8}+3a^{7}z^{7}-4a^{5}z^{7}-13a^{3}z^{7}-3az^{7}+3z^{7}a^{-1}+2a^{8}z^{6}-10a^{6}z^{6}-25a^{4}z^{6}-24a^{2}z^{6}+z^{6}a^{-2}-10z^{6}+a^{9}z^{5}-5a^{7}z^{5}+7a^{3}z^{5}-8az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+16a^{6}z^{4}+43a^{4}z^{4}+32a^{2}z^{4}-3z^{4}a^{-2}+6z^{4}-3a^{9}z^{3}+2a^{7}z^{3}+10a^{5}z^{3}+8a^{3}z^{3}+10az^{3}+7z^{3}a^{-1}+a^{8}z^{2}-13a^{6}z^{2}-30a^{4}z^{2}-22a^{2}z^{2}+2z^{2}a^{-2}-4z^{2}+2a^{9}z-a^{7}z-6a^{5}z-6a^{3}z-5az-2za^{-1}+4a^{6}+9a^{4}+6a^{2}+2$

"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["K11a156"];

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]=

{ $-t^{4}+5t^{3}-11t^{2}+18t-21+18t^{-1}-11t^{-2}+5t^{-3}-t^{-4}$, $q^{3}-3q^{2}+6q-10+13q^{-1}-14q^{-2}+15q^{-3}-12q^{-4}+9q^{-5}-5q^{-6}+2q^{-7}-q^{-8}$ }

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:

(3, -7)

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 $12$ $-56$ $72$ $238$ $42$ $-672$ $-{\frac {3728}{3}}$ $-{\frac {512}{3}}$ $-216$ $288$ $1568$ $2856$ $504$ ${\frac {67711}{10}}$ $-{\frac {1186}{15}}$ ${\frac {43462}{15}}$ ${\frac {673}{6}}$ ${\frac {3711}{10}}$

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 K11a156. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     2   -2 3                   4 1   3 1                 6 2     -4 -1               7 4       3 -3             8 7         -1 -5           7 6           1 -7         5 8             3 -9       4 7               -3 -11     1 5                 4 -13   1 4                   -3 -15   1                     1 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a155 K11a157