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DrawMorseLink

PD Presentation:

X8192 X14,3,15,4 X22,16,13,15 X10,20,11,19 X18,10,19,9 X20,12,21,11 X12,22,7,21 X16,6,17,5 X2738 X4,13,5,14 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

q-2 - 2q-1 + 5 - 6q + 10q2 - 10q3 + 11q4 - 10q5 + 8q6 - 5q7 + 3q8 - q9

A2 (sl(3)) Invariant:

q-6 + q-4 + q-2 + 3 + 2q2 + 5q4 + 4q6 + 3q8 + 4q10 - q12 + 3q14 + q18 + q20 - q22 + q24 - q26

HOMFLY-PT Polynomial:

- a-6 - 4a-6z2 - 4a-6z4 - a-6z6 + a-4z-2 + 5a-4 + 13a-4z2 + 13a-4z4 + 6a-4z6 + a-4z8 - 2a-2z-2 - 8a-2 - 15a-2z2 - 10a-2z4 - 2a-2z6 + z-2 + 4 + 4z2 + z4

Kauffman Polynomial:

a-11z3 - a-10z2 + 3a-10z4 - 3a-9z3 + 5a-9z5 + 2a-8z2 - 9a-8z4 + 7a-8z6 - a-7z + 7a-7z3 - 16a-7z5 + 8a-7z7 + a-6 - 4a-6z2 + 9a-6z4 - 17a-6z6 + 7a-6z8 - 3a-5z + 17a-5z3 - 13a-5z5 - 6a-5z7 + 4a-5z9 - a-4z-2 + 8a-4 - 32a-4z2 + 59a-4z4 - 42a-4z6 + 7a-4z8 + a-4z10 + 2a-3z-1 - 8a-3z + 4a-3z3 + 22a-3z5 - 24a-3z7 + 6a-3z9 - 2a-2z-2 + 12a-2 - 38a-2z2 + 51a-2z4 - 24a-2z6 + a-2z8 + a-2z10 + 2a-1z-1 - 6a-1z - 2a-1z3 + 14a-1z5 - 10a-1z7 + 2a-1z9 - z-2 + 6 - 13z2 + 13z4 - 6z6 + z8

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 19                       1 j = 17                     2   j = 15                   3 1   j = 13                 5 2     j = 11               6 4       j = 9             5 4         j = 7           6 7           j = 5         4 4             j = 3       3 7               j = 1     2 3                 j = -1   1 4                   j = -3   1                     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]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 510]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 510]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 3, 15, 4], X[22, 16, 13, 15], X[10, 20, 11, 19], > X[18, 10, 19, 9], X[20, 12, 21, 11], X[12, 22, 7, 21], X[16, 6, 17, 5], > X[2, 7, 3, 8], X[4, 13, 5, 14], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 8, -11}, {9, -1, 5, -4, 6, -7}, > {10, -2, 3, -8, 11, -5, 4, -6, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-2 2 2 3 4 5 6 7 8 9 5 + q - - - 6 q + 10 q - 10 q + 11 q - 10 q + 8 q - 5 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 -2 2 4 6 8 10 12 14 18 3 + q + q + q + 2 q + 5 q + 4 q + 3 q + 4 q - q + 3 q + q + 20 22 24 26 > q - q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 510]][a, z]
Out[8]=	2 2 2 -6 5 8 -2 1 2 2 4 z 13 z 15 z 4 4 - a + -- - -- + z + ----- - ----- + 4 z - ---- + ----- - ----- + z - 4 2 4 2 2 2 6 4 2 a a a z a z a a a 4 4 4 6 6 6 8 4 z 13 z 10 z z 6 z 2 z z > ---- + ----- - ----- - -- + ---- - ---- + -- 6 4 2 6 4 2 4 a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 510]][a, z]
Out[9]=	-6 8 12 -2 1 2 2 2 z 3 z 8 z 6 z 6 + a + -- + -- - z - ----- - ----- + ---- + --- - -- - --- - --- - --- - 4 2 4 2 2 2 3 a z 7 5 3 a a a a z a z a z a a a 2 2 2 2 2 3 3 3 3 2 z 2 z 4 z 32 z 38 z z 3 z 7 z 17 z > 13 z - --- + ---- - ---- - ----- - ----- + --- - ---- + ---- + ----- + 10 8 6 4 2 11 9 7 5 a a a a a a a a a 3 3 4 4 4 4 4 5 5 4 z 2 z 4 3 z 9 z 9 z 59 z 51 z 5 z 16 z > ---- - ---- + 13 z + ---- - ---- + ---- + ----- + ----- + ---- - ----- - 3 a 10 8 6 4 2 9 7 a a a a a a a a 5 5 5 6 6 6 6 7 7 13 z 22 z 14 z 6 7 z 17 z 42 z 24 z 8 z 6 z > ----- + ----- + ----- - 6 z + ---- - ----- - ----- - ----- + ---- - ---- - 5 3 a 8 6 4 2 7 5 a a a a a a a a 7 7 8 8 8 9 9 9 10 10 24 z 10 z 8 7 z 7 z z 4 z 6 z 2 z z z > ----- - ----- + z + ---- + ---- + -- + ---- + ---- + ---- + --- + --- 3 a 6 4 2 5 3 a 4 2 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 1 1 4 2 q 3 q 3 q 5 7 q + 4 q + ----- + ----- + ---- + ---- + --- + --- + ---- + 4 q t + 5 4 3 3 3 2 2 t t q t q t q t q t t 7 7 2 9 2 9 3 11 3 11 4 13 4 > 6 q t + 7 q t + 5 q t + 4 q t + 6 q t + 4 q t + 5 q t + 13 5 15 5 15 6 17 6 19 7 > 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a510

L11a509 L11a511