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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-6 - 5q-5 + 12q-4 - 18q-3 + 26q-2 - 28q-1 + 29 - 24q + 18q2 - 10q3 + 4q4 - q5

A2 (sl(3)) Invariant:

q-18 - 2q-16 + q-14 + 4q-12 - 2q-10 + 9q-8 + 3q-6 + 4q-4 + 7q-2 - 3 + 7q2 - 4q4 + 2q6 + 3q8 - 4q10 + 2q12 - q14

HOMFLY-PT Polynomial:

- a-2 - 4a-2z2 - 3a-2z4 - a-2z6 + z-2 + 4 + 10z2 + 11z4 + 5z6 + z8 - 2a2z-2 - 4a2 - 7a2z2 - 6a2z4 - 2a2z6 + a4z-2 + a4 + a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 + a-4z2 - 4a-4z4 + 4a-4z6 - a-3z + 7a-3z3 - 12a-3z5 + 9a-3z7 + 2a-2 - 9a-2z2 + 18a-2z4 - 21a-2z6 + 13a-2z8 - 3a-1z + 8a-1z3 - 4a-1z5 - 10a-1z7 + 11a-1z9 - z-2 + 7 - 27z2 + 56z4 - 58z6 + 19z8 + 4z10 + 2az-1 - 4az + 20az5 - 42az7 + 22az9 - 2a2z-2 + 7a2 - 21a2z2 + 50a2z4 - 58a2z6 + 17a2z8 + 4a2z10 + 2a3z-1 - 2a3z + 2a3z3 + 3a3z5 - 18a3z7 + 11a3z9 - a4z-2 + 3a4 - 4a4z2 + 15a4z4 - 24a4z6 + 11a4z8 + 2a5z3 - 8a5z5 + 5a5z7 - a6z4 + a6z6

Khovanov Homology:

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 = 11                       1 j = 9                     3   j = 7                   7 1   j = 5                 11 3     j = 3               13 7       j = 1             16 11         j = -1           14 15           j = -3         12 14             j = -5       8 16               j = -7     4 10                 j = -9   1 8                   j = -11   4                     j = -13 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, 517]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 517]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 4, 15, 3], X[12, 14, 7, 13], X[2, 7, 3, 8], > X[22, 10, 13, 9], X[6, 22, 1, 21], X[20, 16, 21, 15], X[16, 5, 17, 6], > X[18, 11, 19, 12], X[10, 17, 11, 18], X[4, 19, 5, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -11, 8, -6}, {4, -1, 5, -10, 9, -3}, > {3, -2, 7, -8, 10, -9, 11, -7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-6 5 12 18 26 28 2 3 4 5 29 + q - -- + -- - -- + -- - -- - 24 q + 18 q - 10 q + 4 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 -14 4 2 9 3 4 7 2 4 6 -3 + q - --- + q + --- - --- + -- + -- + -- + -- + 7 q - 4 q + 2 q + 16 12 10 8 6 4 2 q q q q q q q 8 10 12 14 > 3 q - 4 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 517]][a, z]
Out[8]=	2 4 2 -2 2 4 -2 2 a a 2 4 z 2 2 4 2 4 - a - 4 a + a + z - ---- + -- + 10 z - ---- - 7 a z + a z + 2 2 2 z z a 4 6 4 3 z 2 4 4 4 6 z 2 6 8 > 11 z - ---- - 6 a z + a z + 5 z - -- - 2 a z + z 2 2 a a
In[9]:=	Kauffman[Link[11, Alternating, 517]][a, z]
Out[9]=	2 4 3 2 2 4 -2 2 a a 2 a 2 a z 3 z 7 + -- + 7 a + 3 a - z - ---- - -- + --- + ---- - -- - --- - 4 a z - 2 2 2 z z 3 a a z z a 2 2 3 3 3 3 2 z 9 z 2 2 4 2 z 7 z 8 z > 2 a z - 27 z + -- - ---- - 21 a z - 4 a z - -- + ---- + ---- + 4 2 5 3 a a a a a 4 4 3 3 5 3 4 4 z 18 z 2 4 4 4 6 4 > 2 a z + 2 a z + 56 z - ---- + ----- + 50 a z + 15 a z - a z + 4 2 a a 5 5 5 6 6 z 12 z 4 z 5 3 5 5 5 6 4 z 21 z > -- - ----- - ---- + 20 a z + 3 a z - 8 a z - 58 z + ---- - ----- - 5 3 a 4 2 a a a a 7 7 2 6 4 6 6 6 9 z 10 z 7 3 7 5 7 > 58 a z - 24 a z + a z + ---- - ----- - 42 a z - 18 a z + 5 a z + 3 a a 8 9 8 13 z 2 8 4 8 11 z 9 3 9 10 > 19 z + ----- + 17 a z + 11 a z + ----- + 22 a z + 11 a z + 4 z + 2 a a 2 10 > 4 a z
In[10]:=	Kh[L][q, t]
Out[10]=	15 1 4 1 8 4 10 8 16 -- + 16 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 12 14 14 3 3 2 5 2 5 3 > ----- + ---- + --- + 11 q t + 13 q t + 7 q t + 11 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a517

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