© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a231 K11a233

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X14,4,15,3 X10,5,11,6 X16,8,17,7 X22,9,1,10 X18,11,19,12 X2,14,3,13 X20,15,21,16 X6,18,7,17 X12,19,13,20 X8,21,9,22

Gauss Code:

{1, -7, 2, -1, 3, -9, 4, -11, 5, -3, 6, -10, 7, -2, 8, -4, 9, -6, 10, -8, 11, -5}

DT (Dowker-Thistlethwaite) Code:

4 14 10 16 22 18 2 20 6 12 8

Alexander Polynomial:

- t-4 + 5t-3 - 14t-2 + 26t-1 - 31 + 26t - 14t2 + 5t3 - t4

Conway Polynomial:

1 - z2 - 4z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{123, -2}

Jones Polynomial:

q-7 - 3q-6 + 7q-5 - 13q-4 + 17q-3 - 19q-2 + 20q-1 - 17 + 13q - 8q2 + 4q3 - q4

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-20 - q-18 + 3q-16 - 2q-14 - 2q-12 + 2q-10 - 4q-8 + 4q-6 - 2q-4 + q-2 + 2 - 2q2 + 4q4 - q6 + q10 - q12

HOMFLY-PT Polynomial:

- a-2 - 2a-2z2 - a-2z4 + 5 + 11z2 + 8z4 + 2z6 - 5a2 - 16a2z2 - 15a2z4 - 6a2z6 - a2z8 + 2a4 + 6a4z2 + 4a4z4 + a4z6

Kauffman Polynomial:

- a-3z + 3a-3z3 - 3a-3z5 + a-3z7 + a-2 - 5a-2z2 + 15a-2z4 - 14a-2z6 + 4a-2z8 - 4a-1z + 9a-1z3 + 3a-1z5 - 13a-1z7 + 5a-1z9 + 5 - 23z2 + 49z4 - 43z6 + 8z8 + 2z10 - 8az + 17az3 + 2az5 - 27az7 + 12az9 + 5a2 - 29a2z2 + 58a2z4 - 56a2z6 + 15a2z8 + 2a2z10 - 9a3z + 29a3z3 - 24a3z5 - 3a3z7 + 7a3z9 + 2a4 - 8a4z2 + 18a4z4 - 21a4z6 + 11a4z8 - 4a5z + 16a5z3 - 17a5z5 + 10a5z7 + 2a6z2 - 5a6z4 + 6a6z6 - 2a7z3 + 3a7z5 - a8z2 + a8z4

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-1, 2}

Khovanov Homology: (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 11232. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 9                       1 j = 7                     3   j = 5                   5 1   j = 3                 8 3     j = 1               9 5       j = -1             11 8         j = -3           9 10           j = -5         8 10             j = -7       5 9               j = -9     2 8                 j = -11   1 5                   j = -13   2                     j = -15 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 232]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 232]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[10, 5, 11, 6], X[16, 8, 17, 7], > X[22, 9, 1, 10], X[18, 11, 19, 12], X[2, 14, 3, 13], X[20, 15, 21, 16], > X[6, 18, 7, 17], X[12, 19, 13, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 232]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -9, 4, -11, 5, -3, 6, -10, 7, -2, 8, -4, 9, -6, 10, > -8, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 232]]
Out[5]=	DTCode[4, 14, 10, 16, 22, 18, 2, 20, 6, 12, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 232]][t]
Out[6]=	-4 5 14 26 2 3 4 -31 - t + -- - -- + -- + 26 t - 14 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 232]][z]
Out[7]=	2 4 6 8 1 - z - 4 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 232]}
In[9]:=	{KnotDet[Knot[11, Alternating, 232]], KnotSignature[Knot[11, Alternating, 232]]}
Out[9]=	{123, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 232]][q]
Out[10]=	-7 3 7 13 17 19 20 2 3 4 -17 + q - -- + -- - -- + -- - -- + -- + 13 q - 8 q + 4 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 232]}
In[12]:=	A2Invariant[Knot[11, Alternating, 232]][q]
Out[12]=	-20 -18 3 2 2 2 4 4 2 -2 2 4 2 + q - q + --- - --- - --- + --- - -- + -- - -- + q - 2 q + 4 q - 16 14 12 10 8 6 4 q q q q q q q 6 10 12 > q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 232]][a, z]
Out[13]=	2 4 -2 2 4 2 2 z 2 2 4 2 4 z 5 - a - 5 a + 2 a + 11 z - ---- - 16 a z + 6 a z + 8 z - -- - 2 2 a a 2 4 4 4 6 2 6 4 6 2 8 > 15 a z + 4 a z + 2 z - 6 a z + a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 232]][a, z]
Out[14]=	2 -2 2 4 z 4 z 3 5 2 5 z 5 + a + 5 a + 2 a - -- - --- - 8 a z - 9 a z - 4 a z - 23 z - ---- - 3 a 2 a a 3 3 2 2 4 2 6 2 8 2 3 z 9 z 3 3 3 > 29 a z - 8 a z + 2 a z - a z + ---- + ---- + 17 a z + 29 a z + 3 a a 4 5 3 7 3 4 15 z 2 4 4 4 6 4 > 16 a z - 2 a z + 49 z + ----- + 58 a z + 18 a z - 5 a z + 2 a 5 5 8 4 3 z 3 z 5 3 5 5 5 7 5 6 > a z - ---- + ---- + 2 a z - 24 a z - 17 a z + 3 a z - 43 z - 3 a a 6 7 7 14 z 2 6 4 6 6 6 z 13 z 7 3 7 > ----- - 56 a z - 21 a z + 6 a z + -- - ----- - 27 a z - 3 a z + 2 3 a a a 8 9 5 7 8 4 z 2 8 4 8 5 z 9 3 9 > 10 a z + 8 z + ---- + 15 a z + 11 a z + ---- + 12 a z + 7 a z + 2 a a 10 2 10 > 2 z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 232]], Vassiliev[3][Knot[11, Alternating, 232]]}
Out[15]=	{-1, 2}
In[16]:=	Kh[Knot[11, Alternating, 232]][q, t]
Out[16]=	10 11 1 2 1 5 2 8 5 9 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 8 10 9 8 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 9 q t + 5 q t + 8 q t + 3 q t + 5 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a232

K11a231 K11a233