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DrawMorseLink

PD Presentation:

X8192 X9,20,10,21 X5,15,6,14 X12,14,7,13 X16,8,17,7 X22,18,13,17 X3,10,4,11 X18,11,19,12 X15,1,16,6 X19,4,20,5 X2,21,3,22

Gauss Code:

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

Jones Polynomial:

q-4 + q-2 + 1 + q + q3 - q4

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- 2a-2 - 4a-2z2 - a-2z4 + z-2 + 7 + 10z2 + 6z4 + z6 - 2a2z-2 - 7a2 - 6a2z2 - a2z4 + a4z-2 + 2a4

Kauffman Polynomial:

- 2a-3z + 9a-3z3 - 6a-3z5 + a-3z7 + 4a-2 - 12a-2z2 + 15a-2z4 - 7a-2z6 + a-2z8 - 6a-1z + 13a-1z3 - 7a-1z5 + a-1z7 - z-2 + 13 - 32z2 + 27z4 - 9z6 + z8 + 2az-1 - 7az + 5az3 - az5 - 2a2z-2 + 13a2 - 24a2z2 + 13a2z4 - 2a2z6 + 2a3z-1 - 3a3z + a3z3 - a4z-2 + 5a4 - 4a4z2 + a4z4

Khovanov Homology:

trqj 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                     j = 5             1 1 1   j = 3             1       j = 1         2 1 1       j = -1       1 3 1         j = -3     1 1 1           j = -5   1 3 1             j = -7     1               j = -9 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]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 428]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 428]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 20, 10, 21], X[5, 15, 6, 14], X[12, 14, 7, 13], > X[16, 8, 17, 7], X[22, 18, 13, 17], X[3, 10, 4, 11], X[18, 11, 19, 12], > X[15, 1, 16, 6], X[19, 4, 20, 5], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -7, 10, -3, 9}, {5, -1, -2, 7, 8, -4}, > {4, 3, -9, -5, 6, -8, -10, 2, 11, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-4 -2 3 4 1 + q + q + q + q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 2 2 3 3 4 4 2 4 6 10 12 4 + --- + --- + --- + -- + -- + -- + -- + 3 q + q + q - q - q 14 12 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 428]][a, z]
Out[8]=	2 4 2 4 2 2 4 -2 2 a a 2 4 z 2 2 4 z 7 - -- - 7 a + 2 a + z - ---- + -- + 10 z - ---- - 6 a z + 6 z - -- - 2 2 2 2 2 a z z a a 2 4 6 > a z + z
In[9]:=	Kauffman[Link[11, NonAlternating, 428]][a, z]
Out[9]=	2 4 3 4 2 4 -2 2 a a 2 a 2 a 2 z 6 z 13 + -- + 13 a + 5 a - z - ---- - -- + --- + ---- - --- - --- - 7 a z - 2 2 2 z z 3 a a z z a 2 3 3 3 2 12 z 2 2 4 2 9 z 13 z 3 > 3 a z - 32 z - ----- - 24 a z - 4 a z + ---- + ----- + 5 a z + 2 3 a a a 4 5 5 3 3 4 15 z 2 4 4 4 6 z 7 z 5 6 > a z + 27 z + ----- + 13 a z + a z - ---- - ---- - a z - 9 z - 2 3 a a a 6 7 7 8 7 z 2 6 z z 8 z > ---- - 2 a z + -- + -- + z + -- 2 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	-3 3 1 1 1 3 1 1 1 1 t q + - + 2 q + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + - + q 9 4 5 3 7 2 5 2 3 2 5 3 q t q q t q t q t q t q t q t q t 2 3 2 5 2 5 3 5 4 9 5 > q t + q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n428

L11n427 L11n429