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

K11n42 K11n44

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,13,6,12 X2837 X18,10,19,9 X20,11,21,12 X13,7,14,6 X10,16,11,15 X22,18,1,17 X14,19,15,20 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 -12 2 18 20 -6 10 22 14 16

Alexander Polynomial:

- 2t-3 + 10t-2 - 20t-1 + 25 - 20t + 10t2 - 2t3

Conway Polynomial:

1 + 2z2 - 2z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{1092, K11a153, K11a224, K11n35, ...}

Determinant and Signature:

{89, 4}

Jones Polynomial:

3q2 - 6q3 + 11q4 - 14q5 + 15q6 - 15q7 + 12q8 - 8q9 + 4q10 - q11

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

{K11n35, ...}

A2 (sl(3)) Invariant:

3q6 - q8 + 4q10 + q12 - 2q14 + 2q16 - 5q18 + q20 - 2q22 + 3q26 - 2q28 + 2q30 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 4a-8 + 7a-8z2 + 3a-8z4 - 7a-6 - 12a-6z2 - 8a-6z4 - 2a-6z6 + 5a-4 + 8a-4z2 + 3a-4z4

Kauffman Polynomial:

- a-13z3 + a-13z5 + 2a-12z2 - 6a-12z4 + 4a-12z6 - 2a-11z + 6a-11z3 - 12a-11z5 + 7a-11z7 + a-10 + 2a-10z2 - 5a-10z4 - 5a-10z6 + 6a-10z8 - 6a-9z + 23a-9z3 - 30a-9z5 + 11a-9z7 + 2a-9z9 + 4a-8 - 10a-8z2 + 16a-8z4 - 20a-8z6 + 11a-8z8 - 8a-7z + 20a-7z3 - 20a-7z5 + 7a-7z7 + 2a-7z9 + 7a-6 - 21a-6z2 + 21a-6z4 - 11a-6z6 + 5a-6z8 - 4a-5z + 4a-5z3 - 3a-5z5 + 3a-5z7 + 5a-4 - 11a-4z2 + 6a-4z4

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

{2, 3}

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=4 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 j = 23                   1 j = 21                 3   j = 19               5 1   j = 17             7 3     j = 15           8 5       j = 13         7 7         j = 11       7 8           j = 9     4 7             j = 7   2 7               j = 5 1 4                 j = 3 3

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, NonAlternating, 43]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 43]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 13, 6, 12], X[2, 8, 3, 7], > X[18, 10, 19, 9], X[20, 11, 21, 12], X[13, 7, 14, 6], X[10, 16, 11, 15], > X[22, 18, 1, 17], X[14, 19, 15, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 43]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, 5, -8, 6, 3, -7, -10, 8, -11, 9, -5, 10, > -6, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 43]]
Out[5]=	DTCode[4, 8, -12, 2, 18, 20, -6, 10, 22, 14, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 43]][t]
Out[6]=	2 10 20 2 3 25 - -- + -- - -- - 20 t + 10 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 43]][z]
Out[7]=	2 4 6 1 + 2 z - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 92], Knot[11, Alternating, 153], Knot[11, Alternating, 224], > Knot[11, NonAlternating, 35], Knot[11, NonAlternating, 43]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 43]], KnotSignature[Knot[11, NonAlternating, 43]]}
Out[9]=	{89, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 43]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 3 q - 6 q + 11 q - 14 q + 15 q - 15 q + 12 q - 8 q + 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 35], Knot[11, NonAlternating, 43]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 43]][q]
Out[12]=	6 8 10 12 14 16 18 20 22 26 28 3 q - q + 4 q + q - 2 q + 2 q - 5 q + q - 2 q + 3 q - 2 q + 30 34 > 2 q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 43]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -10 4 7 5 z 7 z 12 z 8 z 3 z 8 z 3 z 2 z -a + -- - -- + -- - --- + ---- - ----- + ---- + ---- - ---- + ---- - ---- 8 6 4 10 8 6 4 8 6 4 6 a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 43]][a, z]
Out[14]=	2 2 2 2 -10 4 7 5 2 z 6 z 8 z 4 z 2 z 2 z 10 z 21 z a + -- + -- + -- - --- - --- - --- - --- + ---- + ---- - ----- - ----- - 8 6 4 11 9 7 5 12 10 8 6 a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 11 z z 6 z 23 z 20 z 4 z 6 z 5 z 16 z 21 z > ----- - --- + ---- + ----- + ----- + ---- - ---- - ---- + ----- + ----- + 4 13 11 9 7 5 12 10 8 6 a a a a a a a a a a 4 5 5 5 5 5 6 6 6 6 6 z z 12 z 30 z 20 z 3 z 4 z 5 z 20 z 11 z > ---- + --- - ----- - ----- - ----- - ---- + ---- - ---- - ----- - ----- + 4 13 11 9 7 5 12 10 8 6 a a a a a a a a a a 7 7 7 7 8 8 8 9 9 7 z 11 z 7 z 3 z 6 z 11 z 5 z 2 z 2 z > ---- + ----- + ---- + ---- + ---- + ----- + ---- + ---- + ---- 11 9 7 5 10 8 6 9 7 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 43]], Vassiliev[3][Knot[11, NonAlternating, 43]]}
Out[15]=	{2, 3}
In[16]:=	Kh[Knot[11, NonAlternating, 43]][q, t]
Out[16]=	3 5 5 7 7 2 9 2 9 3 11 3 3 q + q + 4 q t + 2 q t + 7 q t + 4 q t + 7 q t + 7 q t + 11 4 13 4 13 5 15 5 15 6 17 6 > 8 q t + 7 q t + 7 q t + 8 q t + 5 q t + 7 q t + 17 7 19 7 19 8 21 8 23 9 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n43

K11n42 K11n44