K11a55

K11a54

K11a56

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a55 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X16,6,17,5 X2837 X18,9,19,10 X20,11,21,12 X22,13,1,14 X6,16,7,15 X14,17,15,18 X10,19,11,20 X12,21,13,22
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -10, 6, -11, 7, -9, 8, -3, 9, -5, 10, -6, 11, -7
Dowker-Thistlethwaite code	4 8 16 2 18 20 22 6 14 10 12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a55/ThurstonBennequinNumber Hyperbolic Volume 11.6846 A-Polynomial See Data:K11a55/A-polynomial

[edit Notes for K11a55'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 K11a55's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-10t^{2}+13t-13+13t^{-1}-10t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 71, -2 }
Jones polynomial	$-q^{4}+2q^{3}-4q^{2}+7q-8+11q^{-1}-11q^{-2}+10q^{-3}-8q^{-4}+5q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-13a^{2}z^{4}-z^{4}a^{-2}+10z^{4}+4a^{4}z^{2}-13a^{2}z^{2}-4z^{2}a^{-2}+15z^{2}+a^{4}-5a^{2}-3a^{-2}+8$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+5az^{9}+2z^{9}a^{-1}+4a^{4}z^{8}+3a^{2}z^{8}+2z^{8}a^{-2}+z^{8}+4a^{5}z^{7}-6a^{3}z^{7}-17az^{7}-6z^{7}a^{-1}+z^{7}a^{-3}+4a^{6}z^{6}-7a^{4}z^{6}-19a^{2}z^{6}-9z^{6}a^{-2}-17z^{6}+3a^{7}z^{5}-3a^{5}z^{5}+4a^{3}z^{5}+14az^{5}-z^{5}a^{-1}-5z^{5}a^{-3}+a^{8}z^{4}-4a^{6}z^{4}+7a^{4}z^{4}+29a^{2}z^{4}+12z^{4}a^{-2}+29z^{4}-4a^{7}z^{3}-2a^{5}z^{3}+a^{3}z^{3}+2az^{3}+10z^{3}a^{-1}+7z^{3}a^{-3}-a^{8}z^{2}-4a^{4}z^{2}-19a^{2}z^{2}-8z^{2}a^{-2}-22z^{2}+a^{7}z+2a^{5}z-a^{3}z-5az-6za^{-1}-3za^{-3}+a^{4}+5a^{2}+3a^{-2}+8$
The A2 invariant	$q^{20}-q^{18}+q^{16}-q^{14}-q^{12}-3q^{8}+2q^{6}-q^{4}+3q^{2}+3+q^{-2}+2q^{-4}-q^{-6}-q^{-10}-q^{-12}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+4q^{106}-2q^{104}-4q^{102}+12q^{100}-18q^{98}+22q^{96}-19q^{94}+8q^{92}+7q^{90}-23q^{88}+37q^{86}-40q^{84}+34q^{82}-19q^{80}-2q^{78}+23q^{76}-36q^{74}+46q^{72}-44q^{70}+33q^{68}-16q^{66}-8q^{64}+30q^{62}-44q^{60}+48q^{58}-35q^{56}+11q^{54}+15q^{52}-37q^{50}+35q^{48}-16q^{46}-16q^{44}+41q^{42}-49q^{40}+26q^{38}+11q^{36}-55q^{34}+81q^{32}-86q^{30}+51q^{28}-2q^{26}-53q^{24}+93q^{22}-101q^{20}+80q^{18}-33q^{16}-17q^{14}+59q^{12}-77q^{10}+73q^{8}-34q^{6}-7q^{4}+46q^{2}-52+39q^{-2}+6q^{-4}-44q^{-6}+68q^{-8}-59q^{-10}+26q^{-12}+25q^{-14}-68q^{-16}+92q^{-18}-78q^{-20}+42q^{-22}+3q^{-24}-47q^{-26}+66q^{-28}-65q^{-30}+45q^{-32}-19q^{-34}-7q^{-36}+22q^{-38}-29q^{-40}+23q^{-42}-16q^{-44}+6q^{-46}-5q^{-50}+4q^{-52}-4q^{-54}+3q^{-56}-q^{-58}+q^{-60}$

Further Quantum Invariants

K11a55 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["K11a55"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-10t^{2}+13t-13+13t^{-1}-10t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 71, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+2q^{3}-4q^{2}+7q-8+11q^{-1}-11q^{-2}+10q^{-3}-8q^{-4}+5q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-13a^{2}z^{4}-z^{4}a^{-2}+10z^{4}+4a^{4}z^{2}-13a^{2}z^{2}-4z^{2}a^{-2}+15z^{2}+a^{4}-5a^{2}-3a^{-2}+8$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+5az^{9}+2z^{9}a^{-1}+4a^{4}z^{8}+3a^{2}z^{8}+2z^{8}a^{-2}+z^{8}+4a^{5}z^{7}-6a^{3}z^{7}-17az^{7}-6z^{7}a^{-1}+z^{7}a^{-3}+4a^{6}z^{6}-7a^{4}z^{6}-19a^{2}z^{6}-9z^{6}a^{-2}-17z^{6}+3a^{7}z^{5}-3a^{5}z^{5}+4a^{3}z^{5}+14az^{5}-z^{5}a^{-1}-5z^{5}a^{-3}+a^{8}z^{4}-4a^{6}z^{4}+7a^{4}z^{4}+29a^{2}z^{4}+12z^{4}a^{-2}+29z^{4}-4a^{7}z^{3}-2a^{5}z^{3}+a^{3}z^{3}+2az^{3}+10z^{3}a^{-1}+7z^{3}a^{-3}-a^{8}z^{2}-4a^{4}z^{2}-19a^{2}z^{2}-8z^{2}a^{-2}-22z^{2}+a^{7}z+2a^{5}z-a^{3}z-5az-6za^{-1}-3za^{-3}+a^{4}+5a^{2}+3a^{-2}+8$

"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["K11a55"];

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]=

{ $-t^{4}+5t^{3}-10t^{2}+13t-13+13t^{-1}-10t^{-2}+5t^{-3}-t^{-4}$, $-q^{4}+2q^{3}-4q^{2}+7q-8+11q^{-1}-11q^{-2}+10q^{-3}-8q^{-4}+5q^{-5}-3q^{-6}+q^{-7}$ }

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:

(2, 1)

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 $8$ $8$ $32$ ${\frac {124}{3}}$ ${\frac {44}{3}}$ $64$ ${\frac {176}{3}}$ $-{\frac {160}{3}}$ $40$ ${\frac {256}{3}}$ $32$ ${\frac {992}{3}}$ ${\frac {352}{3}}$ ${\frac {7231}{15}}$ ${\frac {292}{5}}$ ${\frac {10324}{45}}$ ${\frac {353}{9}}$ ${\frac {271}{15}}$

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 K11a55. 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 9                       1 -1 7                     1   1 5                   3 1   -2 3                 4 1     3 1               4 3       -1 -1             7 4         3 -3           5 5           0 -5         5 6             -1 -7       3 5               2 -9     2 5                 -3 -11   1 3                   2 -13   2                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a54 K11a56