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

K11a42 K11a44

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,6,15,5 X2837 X20,10,21,9 X16,12,17,11 X6,14,7,13 X18,16,19,15 X12,18,13,17 X22,20,1,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 20 16 6 18 12 22 10

Alexander Polynomial:

4t-3 - 15t-2 + 30t-1 - 37 + 30t - 15t2 + 4t3

Conway Polynomial:

1 + 6z2 + 9z4 + 4z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{135, 6}

Jones Polynomial:

q3 - 3q4 + 9q5 - 13q6 + 19q7 - 22q8 + 21q9 - 20q10 + 14q11 - 8q12 + 4q13 - q14

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

{...}

A2 (sl(3)) Invariant:

q10 - 2q12 + 4q14 - q16 + 2q18 + 6q20 - 2q22 + 5q24 - 4q26 - 3q28 - 3q30 - 6q32 + 3q34 + 2q38 + 3q40 - q42 - q44

HOMFLY-PT Polynomial:

- a-14 + 6a-12 + 4a-12z2 - 12a-10 - 17a-10z2 - 6a-10z4 + 7a-8 + 16a-8z2 + 12a-8z4 + 3a-8z6 + a-6 + 3a-6z2 + 3a-6z4 + a-6z6

Kauffman Polynomial:

- a-17z3 + a-17z5 + 3a-16z2 - 6a-16z4 + 4a-16z6 - 3a-15z + 7a-15z3 - 10a-15z5 + 7a-15z7 + a-14 + 8a-14z2 - 11a-14z4 - 2a-14z6 + 7a-14z8 - 15a-13z + 39a-13z3 - 41a-13z5 + 13a-13z7 + 4a-13z9 + 6a-12 - 3a-12z2 + 12a-12z4 - 31a-12z6 + 17a-12z8 + a-12z10 - 27a-11z + 63a-11z3 - 54a-11z5 + 7a-11z7 + 8a-11z9 + 12a-10 - 27a-10z2 + 38a-10z4 - 41a-10z6 + 16a-10z8 + a-10z10 - 15a-9z + 35a-9z3 - 30a-9z5 + 4a-9z7 + 4a-9z9 + 7a-8 - 16a-8z2 + 18a-8z4 - 15a-8z6 + 6a-8z8 + 3a-7z3 - 6a-7z5 + 3a-7z7 - a-6 + 3a-6z2 - 3a-6z4 + a-6z6

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

{6, 12}

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=6 is the signature of 1143. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 29                       1 j = 27                     3   j = 25                   5 1   j = 23                 9 3     j = 21               11 5       j = 19             10 9         j = 17           12 11           j = 15         7 10             j = 13       6 12               j = 11     3 7                 j = 9     6                   j = 7 1 3                     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, 43]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 43]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[2, 8, 3, 7], > X[20, 10, 21, 9], X[16, 12, 17, 11], X[6, 14, 7, 13], X[18, 16, 19, 15], > X[12, 18, 13, 17], X[22, 20, 1, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 43]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -11, 6, -9, 7, -3, 8, -6, 9, -8, 10, > -5, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 43]]
Out[5]=	DTCode[4, 8, 14, 2, 20, 16, 6, 18, 12, 22, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 43]][t]
Out[6]=	4 15 30 2 3 -37 + -- - -- + -- + 30 t - 15 t + 4 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 43]][z]
Out[7]=	2 4 6 1 + 6 z + 9 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 43]}
In[9]:=	{KnotDet[Knot[11, Alternating, 43]], KnotSignature[Knot[11, Alternating, 43]]}
Out[9]=	{135, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 43]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 q - 3 q + 9 q - 13 q + 19 q - 22 q + 21 q - 20 q + 14 q - 8 q + 13 14 > 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 43]}
In[12]:=	A2Invariant[Knot[11, Alternating, 43]][q]
Out[12]=	10 12 14 16 18 20 22 24 26 28 q - 2 q + 4 q - q + 2 q + 6 q - 2 q + 5 q - 4 q - 3 q - 30 32 34 38 40 42 44 > 3 q - 6 q + 3 q + 2 q + 3 q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 43]][a, z]
Out[13]=	2 2 2 2 4 4 -14 6 12 7 -6 4 z 17 z 16 z 3 z 6 z 12 z -a + --- - --- + -- + a + ---- - ----- + ----- + ---- - ---- + ----- + 12 10 8 12 10 8 6 10 8 a a a a a a a a a 4 6 6 3 z 3 z z > ---- + ---- + -- 6 8 6 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 43]][a, z]
Out[14]=	2 2 2 -14 6 12 7 -6 3 z 15 z 27 z 15 z 3 z 8 z 3 z a + --- + --- + -- - a - --- - ---- - ---- - ---- + ---- + ---- - ---- - 12 10 8 15 13 11 9 16 14 12 a a a a a a a a a a 2 2 2 3 3 3 3 3 3 4 27 z 16 z 3 z z 7 z 39 z 63 z 35 z 3 z 6 z > ----- - ----- + ---- - --- + ---- + ----- + ----- + ----- + ---- - ---- - 10 8 6 17 15 13 11 9 7 16 a a a a a a a a a a 4 4 4 4 4 5 5 5 5 11 z 12 z 38 z 18 z 3 z z 10 z 41 z 54 z > ----- + ----- + ----- + ----- - ---- + --- - ----- - ----- - ----- - 14 12 10 8 6 17 15 13 11 a a a a a a a a a 5 5 6 6 6 6 6 6 7 7 30 z 6 z 4 z 2 z 31 z 41 z 15 z z 7 z 13 z > ----- - ---- + ---- - ---- - ----- - ----- - ----- + -- + ---- + ----- + 9 7 16 14 12 10 8 6 15 13 a a a a a a a a a a 7 7 7 8 8 8 8 9 9 9 7 z 4 z 3 z 7 z 17 z 16 z 6 z 4 z 8 z 4 z > ---- + ---- + ---- + ---- + ----- + ----- + ---- + ---- + ---- + ---- + 11 9 7 14 12 10 8 13 11 9 a a a a a a a a a a 10 10 z z > --- + --- 12 10 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 43]], Vassiliev[3][Knot[11, Alternating, 43]]}
Out[15]=	{6, 12}
In[16]:=	Kh[Knot[11, Alternating, 43]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 q + q + 3 q t + 6 q t + 3 q t + 7 q t + 6 q t + 12 q t + 15 4 15 5 17 5 17 6 19 6 19 7 > 7 q t + 10 q t + 12 q t + 11 q t + 10 q t + 9 q t + 21 7 21 8 23 8 23 9 25 9 25 10 > 11 q t + 5 q t + 9 q t + 3 q t + 5 q t + q t + 27 10 29 11 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a43

K11a42 K11a44