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

K11a41 K11a43

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 20 16 6 12 22 10 18

Alexander Polynomial:

t-3 - 9t-2 + 26t-1 - 35 + 26t - 9t2 + t3

Conway Polynomial:

1 - z2 - 3z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{107, 2}

Jones Polynomial:

q-3 - 3q-2 + 7q-1 - 11 + 15q - 17q2 + 17q3 - 15q4 + 11q5 - 6q6 + 3q7 - q8

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

{K11a10, ...}

A2 (sl(3)) Invariant:

q-10 - q-6 + 3q-4 - 2q-2 + 3q2 - 3q4 + 3q6 - 2q8 - 3q14 + 4q16 + 2q22 - q24 - q26

HOMFLY-PT Polynomial:

- a-8 + 3a-6 + 3a-6z2 - 3a-4 - 5a-4z2 - 3a-4z4 + 2a-2 + 3a-2z2 + 2a-2z4 + a-2z6 - 1 - 3z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

a-9z - 2a-9z3 + a-9z5 - a-8 + 4a-8z2 - 6a-8z4 + 3a-8z6 + a-7z - a-7z3 - 4a-7z5 + 4a-7z7 - 3a-6 + 11a-6z2 - 10a-6z4 + 4a-6z8 - a-5z + 5a-5z3 - 7a-5z5 + 2a-5z7 + 3a-5z9 - 3a-4 + 9a-4z2 - 2a-4z4 - 10a-4z6 + 7a-4z8 + a-4z10 - a-3z + 5a-3z3 - 5a-3z5 - 5a-3z7 + 6a-3z9 - 2a-2 + 3a-2z2 + 4a-2z4 - 15a-2z6 + 7a-2z8 + a-2z10 - a-1z + 7a-1z3 - 11a-1z5 + 3a-1z9 - 1 + 4z2 - z4 - 7z6 + 4z8 - az + 6az3 - 8az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

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

{-1, 1}

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 1142. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     2   j = 13                   4 1   j = 11                 7 2     j = 9               8 4       j = 7             9 7         j = 5           8 8           j = 3         7 9             j = 1       5 9               j = -1     2 6                 j = -3   1 5                   j = -5   2                     j = -7 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, 42]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 42]]
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, 9, 21, 10], X[16, 11, 17, 12], X[6, 14, 7, 13], X[12, 15, 13, 16], > X[22, 18, 1, 17], X[10, 19, 11, 20], X[18, 22, 19, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 42]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -8, 7, -3, 8, -6, 9, -11, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 42]]
Out[5]=	DTCode[4, 8, 14, 2, 20, 16, 6, 12, 22, 10, 18]
In[6]:=	alex = Alexander[Knot[11, Alternating, 42]][t]
Out[6]=	-3 9 26 2 3 -35 + t - -- + -- + 26 t - 9 t + t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 42]][z]
Out[7]=	2 4 6 1 - z - 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 42]}
In[9]:=	{KnotDet[Knot[11, Alternating, 42]], KnotSignature[Knot[11, Alternating, 42]]}
Out[9]=	{107, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 42]][q]
Out[10]=	-3 3 7 2 3 4 5 6 7 8 -11 + q - -- + - + 15 q - 17 q + 17 q - 15 q + 11 q - 6 q + 3 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 10], Knot[11, Alternating, 42]}
In[12]:=	A2Invariant[Knot[11, Alternating, 42]][q]
Out[12]=	-10 -6 3 2 2 4 6 8 14 16 22 q - q + -- - -- + 3 q - 3 q + 3 q - 2 q - 3 q + 4 q + 2 q - 4 2 q q 24 26 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 42]][a, z]
Out[13]=	2 2 2 -8 3 3 2 2 2 3 z 5 z 3 z 2 2 4 -1 - a + -- - -- + -- + a - 3 z + ---- - ---- + ---- + a z - 2 z - 6 4 2 6 4 2 a a a a a a 4 4 6 3 z 2 z z > ---- + ---- + -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 42]][a, z]
Out[14]=	2 -8 3 3 2 2 z z z z z 2 4 z -1 - a - -- - -- - -- - a + -- + -- - -- - -- - - - a z + 4 z + ---- + 6 4 2 9 7 5 3 a 8 a a a a a a a a 2 2 2 3 3 3 3 3 11 z 9 z 3 z 2 2 2 z z 5 z 5 z 7 z 3 > ----- + ---- + ---- + 3 a z - ---- - -- + ---- + ---- + ---- + 6 a z - 6 4 2 9 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 4 6 z 10 z 2 z 4 z 2 4 z 4 z 7 z 5 z > z - ---- - ----- - ---- + ---- - 3 a z + -- - ---- - ---- - ---- - 8 6 4 2 9 7 5 3 a a a a a a a a 5 6 6 6 7 7 7 11 z 5 6 3 z 10 z 15 z 2 6 4 z 2 z 5 z > ----- - 8 a z - 7 z + ---- - ----- - ----- + a z + ---- + ---- - ---- + a 8 4 2 7 5 3 a a a a a a 8 8 8 9 9 9 10 10 7 8 4 z 7 z 7 z 3 z 6 z 3 z z z > 3 a z + 4 z + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 42]], Vassiliev[3][Knot[11, Alternating, 42]]}
Out[15]=	{-1, 1}
In[16]:=	Kh[Knot[11, Alternating, 42]][q, t]
Out[16]=	3 1 2 1 5 2 6 5 q 3 9 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 8 q t + 8 q t + 9 q t + 7 q t + 8 q t + 4 q t + 7 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a42

K11a41 K11a43