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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X14,8,15,7 X22,16,17,15 X16,18,5,17 X18,9,19,10 X20,13,21,14 X12,19,13,20 X8,21,9,22 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

q-6 - 3q-5 + 8q-4 - 14q-3 + 20q-2 - 22q-1 + 25 - 20q + 16q2 - 10q3 + 4q4 - q5

A2 (sl(3)) Invariant:

q-18 + q-14 + 4q-12 - 2q-10 + 5q-8 + q-6 + 2q-4 + 9q-2 + 1 + 9q2 - 2q4 + q8 - 4q10 + 2q12 - q14

HOMFLY-PT Polynomial:

- 3a-2 - 4a-2z2 - 3a-2z4 - a-2z6 + z-2 + 10 + 14z2 + 11z4 + 5z6 + z8 - 2a2z-2 - 10a2 - 13a2z2 - 8a2z4 - 2a2z6 + a4z-2 + 3a4 + 3a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 - 4a-4z4 + 4a-4z6 - 3a-3z + 8a-3z3 - 14a-3z5 + 9a-3z7 + 5a-2 - 12a-2z2 + 21a-2z4 - 23a-2z6 + 12a-2z8 - 10a-1z + 20a-1z3 - 14a-1z5 - 4a-1z7 + 8a-1z9 - z-2 + 15 - 38z2 + 61z4 - 55z6 + 19z8 + 2z10 + 2az-1 - 13az + 15az3 + az5 - 20az7 + 13az9 - 2a2z-2 + 13a2 - 31a2z2 + 43a2z4 - 38a2z6 + 12a2z8 + 2a2z10 + 2a3z-1 - 7a3z + 9a3z3 - 7a3z5 - 4a3z7 + 5a3z9 - a4z-2 + 3a4 - 2a4z2 + 4a4z4 - 9a4z6 + 5a4z8 - a5z + 5a5z3 - 7a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + 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                 9 3     j = 3               11 7       j = 1             14 9         j = -1           10 13           j = -3         10 12             j = -5       6 12               j = -7     2 8                 j = -9   1 6                   j = -11   2                     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, 476]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 476]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[14, 8, 15, 7], X[22, 16, 17, 15], > X[16, 18, 5, 17], X[18, 9, 19, 10], X[20, 13, 21, 14], X[12, 19, 13, 20], > X[8, 21, 9, 22], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {5, -6, 8, -7, 9, -4}, > {10, -1, 3, -9, 6, -2, 11, -8, 7, -3, 4, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-6 3 8 14 20 22 2 3 4 5 25 + q - -- + -- - -- + -- - -- - 20 q + 16 q - 10 q + 4 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -14 4 2 5 -6 2 9 2 4 8 10 1 + q + q + --- - --- + -- + q + -- + -- + 9 q - 2 q + q - 4 q + 12 10 8 4 2 q q q q q 12 14 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 476]][a, z]
Out[8]=	2 4 2 3 2 4 -2 2 a a 2 4 z 2 2 4 2 10 - -- - 10 a + 3 a + z - ---- + -- + 14 z - ---- - 13 a z + 3 a z + 2 2 2 2 a z z a 4 6 4 3 z 2 4 4 4 6 z 2 6 8 > 11 z - ---- - 8 a z + a z + 5 z - -- - 2 a z + z 2 2 a a
In[9]:=	Kauffman[Link[11, Alternating, 476]][a, z]
Out[9]=	2 4 3 5 2 4 6 -2 2 a a 2 a 2 a 3 z 10 z 15 + -- + 13 a + 3 a - a - z - ---- - -- + --- + ---- - --- - ---- - 2 2 2 z z 3 a a z z a 2 3 5 2 12 z 2 2 4 2 6 2 > 13 a z - 7 a z - a z - 38 z - ----- - 31 a z - 2 a z + 3 a z - 2 a 3 3 3 4 4 z 8 z 20 z 3 3 3 5 3 4 4 z 21 z > -- + ---- + ----- + 15 a z + 9 a z + 5 a z + 61 z - ---- + ----- + 5 3 a 4 2 a a a a 5 5 5 2 4 4 4 6 4 z 14 z 14 z 5 3 5 > 43 a z + 4 a z - 3 a z + -- - ----- - ----- + a z - 7 a z - 5 3 a a a 6 6 7 7 5 5 6 4 z 23 z 2 6 4 6 6 6 9 z 4 z > 7 a z - 55 z + ---- - ----- - 38 a z - 9 a z + a z + ---- - ---- - 4 2 3 a a a a 8 9 7 3 7 5 7 8 12 z 2 8 4 8 8 z > 20 a z - 4 a z + 3 a z + 19 z + ----- + 12 a z + 5 a z + ---- + 2 a a 9 3 9 10 2 10 > 13 a z + 5 a z + 2 z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	13 1 2 1 6 2 8 6 12 -- + 14 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 10 12 10 3 3 2 5 2 5 3 > ----- + ---- + --- + 9 q t + 11 q t + 7 q t + 9 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 L11a476

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