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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9 - 3q-8 + 8q-7 - 12q-6 + 19q-5 - 20q-4 + 22q-3 - 19q-2 + 14q-1 - 9 + 4q - q2

A2 (sl(3)) Invariant:

q-28 + q-26 + q-24 + 7q-22 + 3q-20 + 5q-18 + 9q-16 + q-14 + 6q-12 - q-10 - q-8 - 6q-4 + 3q-2 - 2 - q2 + 2q4 - q6

HOMFLY-PT Polynomial:

- z2 - z4 - a2z-2 - 4a2 - 2a2z2 + a2z4 + a2z6 + 4a4z-2 + 14a4 + 16a4z2 + 8a4z4 + 2a4z6 - 5a6z-2 - 12a6 - 9a6z2 - 3a6z4 + 2a8z-2 + 2a8 + a8z2

Kauffman Polynomial:

- a-1z3 + a-1z5 + z2 - 5z4 + 4z6 + az-1 - 4az + 8az3 - 13az5 + 8az7 - a2z-2 + 3a2 - 6a2z2 + 8a2z4 - 13a2z6 + 9a2z8 + 5a3z-1 - 19a3z + 41a3z3 - 37a3z5 + 8a3z7 + 5a3z9 - 4a4z-2 + 15a4 - 38a4z2 + 62a4z4 - 55a4z6 + 20a4z8 + a4z10 + 9a5z-1 - 33a5z + 53a5z3 - 34a5z5 - 3a5z7 + 9a5z9 - 5a6z-2 + 20a6 - 45a6z2 + 62a6z4 - 50a6z6 + 16a6z8 + a6z10 + 5a7z-1 - 18a7z + 25a7z3 - 18a7z5 + 4a7z9 - 2a8z-2 + 8a8 - 11a8z2 + 10a8z4 - 11a8z6 + 5a8z8 + 4a9z3 - 7a9z5 + 3a9z7 - a10 + 3a10z2 - 3a10z4 + a10z6

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                       1 j = 3                     3   j = 1                   6 1   j = -1                 8 3     j = -3               12 7       j = -5             10 7         j = -7           10 12           j = -9         9 10             j = -11       4 11               j = -13     4 8                 j = -15   1 6                   j = -17   2                     j = -19 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, 404]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 404]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[10, 13, 5, 14], X[8, 17, 9, 18], > X[14, 7, 15, 8], X[18, 9, 19, 10], X[22, 20, 11, 19], X[20, 16, 21, 15], > X[16, 22, 17, 21], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 5, -4, 6, -3}, > {11, -2, 3, -5, 8, -9, 4, -6, 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-9 3 8 12 19 20 22 19 14 2 -9 + q - -- + -- - -- + -- - -- + -- - -- + -- + 4 q - q 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 -24 7 3 5 9 -14 6 -10 -8 -2 + q + q + q + --- + --- + --- + --- + q + --- - q - q - 22 20 18 16 12 q q q q q 6 3 2 4 6 > -- + -- - q + 2 q - q 4 2 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 404]][a, z]
Out[8]=	2 4 6 8 2 4 6 8 a 4 a 5 a 2 a 2 2 2 -4 a + 14 a - 12 a + 2 a - -- + ---- - ---- + ---- - z - 2 a z + 2 2 2 2 z z z z 4 2 6 2 8 2 4 2 4 4 4 6 4 2 6 > 16 a z - 9 a z + a z - z + a z + 8 a z - 3 a z + a z + 4 6 > 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 404]][a, z]
Out[9]=	2 4 6 8 3 5 2 4 6 8 10 a 4 a 5 a 2 a a 5 a 9 a 3 a + 15 a + 20 a + 8 a - a - -- - ---- - ---- - ---- + - + ---- + ---- + 2 2 2 2 z z z z z z z 7 5 a 3 5 7 2 2 2 4 2 > ---- - 4 a z - 19 a z - 33 a z - 18 a z + z - 6 a z - 38 a z - z 3 6 2 8 2 10 2 z 3 3 3 5 3 > 45 a z - 11 a z + 3 a z - -- + 8 a z + 41 a z + 53 a z + a 7 3 9 3 4 2 4 4 4 6 4 8 4 > 25 a z + 4 a z - 5 z + 8 a z + 62 a z + 62 a z + 10 a z - 5 10 4 z 5 3 5 5 5 7 5 9 5 6 > 3 a z + -- - 13 a z - 37 a z - 34 a z - 18 a z - 7 a z + 4 z - a 2 6 4 6 6 6 8 6 10 6 7 3 7 > 13 a z - 55 a z - 50 a z - 11 a z + a z + 8 a z + 8 a z - 5 7 9 7 2 8 4 8 6 8 8 8 3 9 > 3 a z + 3 a z + 9 a z + 20 a z + 16 a z + 5 a z + 5 a z + 5 9 7 9 4 10 6 10 > 9 a z + 4 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 8 1 2 1 6 4 8 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 11 9 10 10 12 10 7 12 3 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a404

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