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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9 - 4q-8 + 10q-7 - 18q-6 + 25q-5 - 28q-4 + 31q-3 - 25q-2 + 20q-1 - 12 + 5q - q2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- z4 + a2z-2 + a2 - a2z2 + a2z6 - 2a4z-2 + a4 + 6a4z2 + 5a4z4 + 2a4z6 + a6z-2 - 3a6 - 5a6z2 - 3a6z4 + a8 + a8z2

Kauffman Polynomial:

a-1z5 - 3z4 + 5z6 - az + 4az3 - 14az5 + 12az7 + a2z-2 - 5a2z2 + 16a2z4 - 27a2z6 + 17a2z8 - 2a3z-1 - 3a3z + 17a3z3 - 22a3z5 - 4a3z7 + 12a3z9 + 2a4z-2 + 3a4 - 21a4z2 + 63a4z4 - 79a4z6 + 30a4z8 + 3a4z10 - 2a5z-1 - 7a5z + 25a5z3 - 13a5z5 - 26a5z7 + 20a5z9 + a6z-2 + 4a6 - 24a6z2 + 59a6z4 - 64a6z6 + 21a6z8 + 3a6z10 - 7a7z + 18a7z3 - 14a7z5 - 6a7z7 + 8a7z9 + 2a8 - 7a8z2 + 13a8z4 - 16a8z6 + 8a8z8 - 2a9z + 6a9z3 - 8a9z5 + 4a9z7 + a10z2 - 2a10z4 + 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                     4   j = 1                   8 1   j = -1                 12 4     j = -3               14 9       j = -5             17 11         j = -7           13 16           j = -9         12 15             j = -11       7 14               j = -13     3 11                 j = -15   1 7                   j = -17   3                     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, 412]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 412]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 13, 19, 14], X[22, 17, 11, 18], > X[16, 7, 17, 8], X[8, 22, 9, 21], X[14, 10, 15, 9], X[20, 16, 21, 15], > X[10, 19, 5, 20], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 5, -6, 7, -9}, > {11, -2, 3, -7, 8, -5, 4, -3, 9, -8, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-9 4 10 18 25 28 31 25 20 2 -12 + q - -- + -- - -- + -- - -- + -- - -- + -- + 5 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 2 4 4 -18 7 -14 10 3 5 6 4 -3 + q - --- + --- - --- - q + --- - q + --- + --- + -- + -- - -- + 24 22 20 16 12 10 8 6 4 q q q q q q q q q 6 2 4 6 > -- - 2 q + 3 q - q 2 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 412]][a, z]
Out[8]=	2 4 6 2 4 6 8 a 2 a a 2 2 4 2 6 2 8 2 4 a + a - 3 a + a + -- - ---- + -- - a z + 6 a z - 5 a z + a z - z + 2 2 2 z z z 4 4 6 4 2 6 4 6 > 5 a z - 3 a z + a z + 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 412]][a, z]
Out[9]=	2 4 6 3 5 4 6 8 a 2 a a 2 a 2 a 3 5 3 a + 4 a + 2 a + -- + ---- + -- - ---- - ---- - a z - 3 a z - 7 a z - 2 2 2 z z z z z 7 9 2 2 4 2 6 2 8 2 10 2 > 7 a z - 2 a z - 5 a z - 21 a z - 24 a z - 7 a z + a z + 3 3 3 5 3 7 3 9 3 4 2 4 > 4 a z + 17 a z + 25 a z + 18 a z + 6 a z - 3 z + 16 a z + 5 4 4 6 4 8 4 10 4 z 5 3 5 > 63 a z + 59 a z + 13 a z - 2 a z + -- - 14 a z - 22 a z - a 5 5 7 5 9 5 6 2 6 4 6 6 6 > 13 a z - 14 a z - 8 a z + 5 z - 27 a z - 79 a z - 64 a z - 8 6 10 6 7 3 7 5 7 7 7 9 7 > 16 a z + a z + 12 a z - 4 a z - 26 a z - 6 a z + 4 a z + 2 8 4 8 6 8 8 8 3 9 5 9 7 9 > 17 a z + 30 a z + 21 a z + 8 a z + 12 a z + 20 a z + 8 a z + 4 10 6 10 > 3 a z + 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 12 1 3 1 7 3 11 7 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 14 12 15 13 16 17 11 14 4 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 > 8 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a412

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