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

K11a22 K11a24

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X12,5,13,6 X2837 X18,10,19,9 X14,11,15,12 X6,13,7,14 X22,16,1,15 X20,18,21,17 X10,20,11,19 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 12 2 18 14 6 22 20 10 16

Alexander Polynomial:

2t-3 - 10t-2 + 24t-1 - 31 + 24t - 10t2 + 2t3

Conway Polynomial:

1 + 2z2 + 2z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{10117, K11a111, ...}

Determinant and Signature:

{103, 2}

Jones Polynomial:

- q-2 + 3q-1 - 6 + 11q - 14q2 + 17q3 - 16q4 + 14q5 - 11q6 + 6q7 - 3q8 + q9

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

{...}

A2 (sl(3)) Invariant:

- q-6 + q-4 - q-2 - 1 + 4q2 - 2q4 + 3q6 + 2q8 + 3q12 - 3q14 + q16 - 2q18 - 3q20 + 2q22 - q24 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 3a-6 - 4a-6z2 - 2a-6z4 + a-4 + 2a-4z2 + 2a-4z4 + a-4z6 + 3a-2 + 5a-2z2 + 3a-2z4 + a-2z6 - 1 - 2z2 - z4

Kauffman Polynomial:

2a-10z2 - 3a-10z4 + a-10z6 - 3a-9z + 8a-9z3 - 9a-9z5 + 3a-9z7 + a-8 - a-8z2 + 4a-8z4 - 9a-8z6 + 4a-8z8 - 8a-7z + 17a-7z3 - 13a-7z5 - a-7z7 + 3a-7z9 + 3a-6 - 10a-6z2 + 20a-6z4 - 21a-6z6 + 7a-6z8 + a-6z10 - 5a-5z + 11a-5z3 - 4a-5z5 - 7a-5z7 + 6a-5z9 + a-4 - 4a-4z2 + 12a-4z4 - 15a-4z6 + 7a-4z8 + a-4z10 + a-3z + 2a-3z3 - 5a-3z5 + a-3z7 + 3a-3z9 - 3a-2 + 7a-2z2 - 7a-2z4 - a-2z6 + 4a-2z8 + 2a-1z - 2a-1z3 - 4a-1z5 + 4a-1z7 - 1 + 4z2 - 6z4 + 3z6 + az - 2az3 + az5

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

{2, 1}

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 1123. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                       1 j = 17                     2   j = 15                   4 1   j = 13                 7 2     j = 11               7 4       j = 9             9 7         j = 7           8 7           j = 5         6 9             j = 3       5 8               j = 1     2 7                 j = -1   1 4                   j = -3   2                     j = -5 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, 23]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 23]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[18, 10, 19, 9], X[14, 11, 15, 12], X[6, 13, 7, 14], X[22, 16, 1, 15], > X[20, 18, 21, 17], X[10, 20, 11, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 23]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -6, 8, -11, 9, -5, 10, > -9, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 23]]
Out[5]=	DTCode[4, 8, 12, 2, 18, 14, 6, 22, 20, 10, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 23]][t]
Out[6]=	2 10 24 2 3 -31 + -- - -- + -- + 24 t - 10 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 23]][z]
Out[7]=	2 4 6 1 + 2 z + 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 117], Knot[11, Alternating, 23], Knot[11, Alternating, 111]}
In[9]:=	{KnotDet[Knot[11, Alternating, 23]], KnotSignature[Knot[11, Alternating, 23]]}
Out[9]=	{103, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 23]][q]
Out[10]=	-2 3 2 3 4 5 6 7 8 9 -6 - q + - + 11 q - 14 q + 17 q - 16 q + 14 q - 11 q + 6 q - 3 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 23]}
In[12]:=	A2Invariant[Knot[11, Alternating, 23]][q]
Out[12]=	-6 -4 -2 2 4 6 8 12 14 16 -1 - q + q - q + 4 q - 2 q + 3 q + 2 q + 3 q - 3 q + q - 18 20 22 24 28 > 2 q - 3 q + 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 23]][a, z]
Out[13]=	2 2 2 2 4 4 -8 3 -4 3 2 z 4 z 2 z 5 z 4 2 z 2 z -1 + a - -- + a + -- - 2 z + -- - ---- + ---- + ---- - z - ---- + ---- + 6 2 8 6 4 2 6 4 a a a a a a a a 4 6 6 3 z z z > ---- + -- + -- 2 4 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 23]][a, z]
Out[14]=	2 -8 3 -4 3 3 z 8 z 5 z z 2 z 2 2 z -1 + a + -- + a - -- - --- - --- - --- + -- + --- + a z + 4 z + ---- - 6 2 9 7 5 3 a 10 a a a a a a a 2 2 2 2 3 3 3 3 3 z 10 z 4 z 7 z 8 z 17 z 11 z 2 z 2 z 3 > -- - ----- - ---- + ---- + ---- + ----- + ----- + ---- - ---- - 2 a z - 8 6 4 2 9 7 5 3 a a a a a a a a a 4 4 4 4 4 5 5 5 5 4 3 z 4 z 20 z 12 z 7 z 9 z 13 z 4 z 5 z > 6 z - ---- + ---- + ----- + ----- - ---- - ---- - ----- - ---- - ---- - 10 8 6 4 2 9 7 5 3 a a a a a a a a a 5 6 6 6 6 6 7 7 7 4 z 5 6 z 9 z 21 z 15 z z 3 z z 7 z > ---- + a z + 3 z + --- - ---- - ----- - ----- - -- + ---- - -- - ---- + a 10 8 6 4 2 9 7 5 a a a a a a a a 7 7 8 8 8 8 9 9 9 10 10 z 4 z 4 z 7 z 7 z 4 z 3 z 6 z 3 z z z > -- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- 3 a 8 6 4 2 7 5 3 6 4 a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 23]], Vassiliev[3][Knot[11, Alternating, 23]]}
Out[15]=	{2, 1}
In[16]:=	Kh[Knot[11, Alternating, 23]][q, t]
Out[16]=	3 1 2 1 4 2 q 3 5 5 2 7 q + 5 q + ----- + ----- + ---- + --- + --- + 8 q t + 6 q t + 9 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 8 q t + 7 q t + 9 q t + 7 q t + 7 q t + 4 q t + 7 q t + 13 6 15 6 15 7 17 7 19 8 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a23

K11a22 K11a24