© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a348 K11a350

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X18,4,19,3 X16,5,17,6 X12,8,13,7 X4,10,5,9 X2,11,3,12 X22,14,1,13 X20,16,21,15 X10,18,11,17 X8,19,9,20 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 18 16 12 4 2 22 20 10 8 14

Alexander Polynomial:

2t-3 - 14t-2 + 37t-1 - 49 + 37t - 14t2 + 2t3

Conway Polynomial:

1 - z2 - 2z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{155, 2}

Jones Polynomial:

- q-2 + 5q-1 - 10 + 17q - 22q2 + 25q3 - 25q4 + 21q5 - 15q6 + 9q7 - 4q8 + q9

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-6 + 3q-4 - q-2 - 1 + 6q2 - 4q4 + 3q6 - 3q10 + 2q12 - 5q14 + 4q16 - 2q20 + 4q22 - 2q24 - q26 + q28

HOMFLY-PT Polynomial:

a-8z2 + a-6 - a-6z2 - 2a-6z4 - 2a-4 - 2a-4z2 + a-4z6 + a-2 + a-2z2 + a-2z4 + a-2z6 + 1 - z4

Kauffman Polynomial:

- 2a-10z4 + a-10z6 + 3a-9z3 - 9a-9z5 + 4a-9z7 - 6a-8z2 + 17a-8z4 - 21a-8z6 + 8a-8z8 + 2a-7z - 10a-7z3 + 22a-7z5 - 23a-7z7 + 9a-7z9 - a-6 - 8a-6z2 + 35a-6z4 - 32a-6z6 + 5a-6z8 + 4a-6z10 + 3a-5z - 19a-5z3 + 49a-5z5 - 49a-5z7 + 19a-5z9 - 2a-4 + 21a-4z4 - 30a-4z6 + 9a-4z8 + 4a-4z10 + a-3z - 4a-3z3 + 4a-3z5 - 12a-3z7 + 10a-3z9 - a-2 + 2a-2z2 - 15a-2z6 + 12a-2z8 + 2a-1z3 - 13a-1z5 + 10a-1z7 + 1 - 5z4 + 5z6 + az5

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-1, -2}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=2 is the signature of 11349. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                       1 j = 17                     3   j = 15                   6 1   j = 13                 9 3     j = 11               12 6       j = 9             13 9         j = 7           12 12           j = 5         10 13             j = 3       7 12               j = 1     4 11                 j = -1   1 6                   j = -3   4                     j = -5 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]:=	Show[DrawMorseLink[Knot[11, Alternating, 349]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 349]]
Out[3]=	PD[X[6, 2, 7, 1], X[18, 4, 19, 3], X[16, 5, 17, 6], X[12, 8, 13, 7], > X[4, 10, 5, 9], X[2, 11, 3, 12], X[22, 14, 1, 13], X[20, 16, 21, 15], > X[10, 18, 11, 17], X[8, 19, 9, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 349]]
Out[4]=	GaussCode[1, -6, 2, -5, 3, -1, 4, -10, 5, -9, 6, -4, 7, -11, 8, -3, 9, -2, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 349]]
Out[5]=	DTCode[6, 18, 16, 12, 4, 2, 22, 20, 10, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 349]][t]
Out[6]=	2 14 37 2 3 -49 + -- - -- + -- + 37 t - 14 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 349]][z]
Out[7]=	2 4 6 1 - z - 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 349]}
In[9]:=	{KnotDet[Knot[11, Alternating, 349]], KnotSignature[Knot[11, Alternating, 349]]}
Out[9]=	{155, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 349]][q]
Out[10]=	-2 5 2 3 4 5 6 7 8 9 -10 - q + - + 17 q - 22 q + 25 q - 25 q + 21 q - 15 q + 9 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 349]}
In[12]:=	A2Invariant[Knot[11, Alternating, 349]][q]
Out[12]=	-6 3 -2 2 4 6 10 12 14 16 -1 - q + -- - q + 6 q - 4 q + 3 q - 3 q + 2 q - 5 q + 4 q - 4 q 20 22 24 26 28 > 2 q + 4 q - 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 349]][a, z]
Out[13]=	2 2 2 2 4 4 6 6 -6 2 -2 z z 2 z z 4 2 z z z z 1 + a - -- + a + -- - -- - ---- + -- - z - ---- + -- + -- + -- 4 8 6 4 2 6 2 4 2 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 349]][a, z]
Out[14]=	2 2 2 3 3 -6 2 -2 2 z 3 z z 6 z 8 z 2 z 3 z 10 z 1 - a - -- - a + --- + --- + -- - ---- - ---- + ---- + ---- - ----- - 4 7 5 3 8 6 2 9 7 a a a a a a a a a 3 3 3 4 4 4 4 5 5 19 z 4 z 2 z 4 2 z 17 z 35 z 21 z 9 z 22 z > ----- - ---- + ---- - 5 z - ---- + ----- + ----- + ----- - ---- + ----- + 5 3 a 10 8 6 4 9 7 a a a a a a a a 5 5 5 6 6 6 6 6 49 z 4 z 13 z 5 6 z 21 z 32 z 30 z 15 z > ----- + ---- - ----- + a z + 5 z + --- - ----- - ----- - ----- - ----- + 5 3 a 10 8 6 4 2 a a a a a a a 7 7 7 7 7 8 8 8 8 9 4 z 23 z 49 z 12 z 10 z 8 z 5 z 9 z 12 z 9 z > ---- - ----- - ----- - ----- + ----- + ---- + ---- + ---- + ----- + ---- + 9 7 5 3 a 8 6 4 2 7 a a a a a a a a a 9 9 10 10 19 z 10 z 4 z 4 z > ----- + ----- + ----- + ----- 5 3 6 4 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 349]], Vassiliev[3][Knot[11, Alternating, 349]]}
Out[15]=	{-1, -2}
In[16]:=	Kh[Knot[11, Alternating, 349]][q, t]
Out[16]=	3 1 4 1 6 4 q 3 5 5 2 11 q + 7 q + ----- + ----- + ---- + --- + --- + 12 q t + 10 q t + 13 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 > 12 q t + 12 q t + 13 q t + 9 q t + 12 q t + 6 q t + 13 5 13 6 15 6 15 7 17 7 19 8 > 9 q t + 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a349

K11a348 K11a350