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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X15,1,16,4 X5,12,6,13 X3849 X9,16,10,17 X17,19,18,22 X11,20,12,21 X19,10,20,11 X21,5,22,18 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- q-8 + 2q-7 - 4q-6 + 6q-5 - 8q-4 + 9q-3 - 6q-2 + 7q-1 - 3 + 2q

A2 (sl(3)) Invariant:

- q-24 - 2q-20 - 2q-18 - 3q-14 + q-12 + 2q-10 + 5q-8 + 8q-6 + 5q-4 + 7q-2 + 3 + 2q2 + 2q4

HOMFLY-PT Polynomial:

2z-2 + 4 + 2z2 - 5a2z-2 - 11a2 - 10a2z2 - 3a2z4 + 4a4z-2 + 10a4 + 11a4z2 + 5a4z4 + a4z6 - a6z-2 - 3a6 - 3a6z2 - a6z4

Kauffman Polynomial:

- 2z-2 + 7 - 8z2 + 3z4 + 5az-1 - 10az + 3az3 - az5 + az7 - 5a2z-2 + 17a2 - 32a2z2 + 23a2z4 - 8a2z6 + 2a2z8 + 9a3z-1 - 26a3z + 26a3z3 - 8a3z5 + a3z9 - 4a4z-2 + 16a4 - 34a4z2 + 39a4z4 - 20a4z6 + 5a4z8 + 5a5z-1 - 22a5z + 35a5z3 - 17a5z5 + 2a5z7 + a5z9 - a6z-2 + 5a6 - 9a6z2 + 14a6z4 - 10a6z6 + 3a6z8 + a7z-1 - 5a7z + 9a7z3 - 9a7z5 + 3a7z7 + a8z2 - 5a8z4 + 2a8z6 + a9z - 3a9z3 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                   2 j = 1                 1   j = -1               6 2   j = -3             4 5     j = -5           5 2       j = -7         3 4         j = -9       3 5           j = -11     1 3             j = -13   1 3               j = -15   1                 j = -17 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, 386]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 386]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[15, 1, 16, 4], X[5, 12, 6, 13], > X[3, 8, 4, 9], X[9, 16, 10, 17], X[17, 19, 18, 22], X[11, 20, 12, 21], > X[19, 10, 20, 11], X[21, 5, 22, 18], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -5, 3}, {-9, 8, -10, 7}, > {-4, -1, 2, 5, -6, 9, -8, 4, 11, -2, -3, 6, -7, 10}]
In[6]:=	Jones[L][q]
Out[6]=	-8 2 4 6 8 9 6 7 -3 - q + -- - -- + -- - -- + -- - -- + - + 2 q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 2 3 -12 2 5 8 5 7 2 4 3 - q - --- - --- - --- + q + --- + -- + -- + -- + -- + 2 q + 2 q 20 18 14 10 8 6 4 2 q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 386]][a, z]
Out[8]=	2 4 6 2 4 6 2 5 a 4 a a 2 2 2 4 2 4 - 11 a + 10 a - 3 a + -- - ---- + ---- - -- + 2 z - 10 a z + 11 a z - 2 2 2 2 z z z z 6 2 2 4 4 4 6 4 4 6 > 3 a z - 3 a z + 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 386]][a, z]
Out[9]=	2 4 6 3 5 7 2 4 6 2 5 a 4 a a 5 a 9 a 5 a a 7 + 17 a + 16 a + 5 a - -- - ---- - ---- - -- + --- + ---- + ---- + -- - 2 2 2 2 z z z z z z z z 3 5 7 9 2 2 2 4 2 > 10 a z - 26 a z - 22 a z - 5 a z + a z - 8 z - 32 a z - 34 a z - 6 2 8 2 3 3 3 5 3 7 3 9 3 4 > 9 a z + a z + 3 a z + 26 a z + 35 a z + 9 a z - 3 a z + 3 z + 2 4 4 4 6 4 8 4 5 3 5 5 5 > 23 a z + 39 a z + 14 a z - 5 a z - a z - 8 a z - 17 a z - 7 5 9 5 2 6 4 6 6 6 8 6 7 > 9 a z + a z - 8 a z - 20 a z - 10 a z + 2 a z + a z + 5 7 7 7 2 8 4 8 6 8 3 9 5 9 > 2 a z + 3 a z + 2 a z + 5 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 6 1 1 1 3 1 3 3 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 3 4 5 2 4 2 t 3 2 > ----- + ----- + ----- + ---- + ---- + --- + q t + 2 q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n386

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