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

K11a40 K11a42

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 18 20 16 6 22 12 10

Alexander Polynomial:

2t-3 - 11t-2 + 27t-1 - 35 + 27t - 11t2 + 2t3

Conway Polynomial:

1 + z2 + z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{10121, K11a183, K11a198, K11a331, ...}

Determinant and Signature:

{115, 2}

Jones Polynomial:

- q-2 + 4q-1 - 8 + 13q - 16q2 + 19q3 - 18q4 + 15q5 - 11q6 + 6q7 - 3q8 + q9

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

{K11a183, ...}

A2 (sl(3)) Invariant:

- q-6 + 2q-4 - q-2 - 1 + 4q2 - 3q4 + 3q6 + q8 + 3q12 - 3q14 + 2q16 - 2q18 - 3q20 + 2q22 - q24 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 3a-6 - 4a-6z2 - 2a-6z4 + 2a-4 + 3a-4z2 + 2a-4z4 + a-4z6 + a-2 + 2a-2z2 + 2a-2z4 + a-2z6 - z2 - z4

Kauffman Polynomial:

2a-10z2 - 3a-10z4 + a-10z6 - 3a-9z + 8a-9z3 - 9a-9z5 + 3a-9z7 + a-8 - 2a-8z2 + 5a-8z4 - 9a-8z6 + 4a-8z8 - 8a-7z + 19a-7z3 - 15a-7z5 + 3a-7z9 + 3a-6 - 13a-6z2 + 28a-6z4 - 27a-6z6 + 9a-6z8 + a-6z10 - 7a-5z + 20a-5z3 - 12a-5z5 - 6a-5z7 + 7a-5z9 + 2a-4 - 10a-4z2 + 26a-4z4 - 29a-4z6 + 12a-4z8 + a-4z10 - 3a-3z + 14a-3z3 - 18a-3z5 + 4a-3z7 + 4a-3z9 - a-2 + a-2z2 - 8a-2z6 + 7a-2z8 - a-1z + 4a-1z3 - 11a-1z5 + 7a-1z7 + 2z2 - 6z4 + 4z6 - az3 + az5

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

{1, 0}

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 1141. 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                     2   j = 15                   4 1   j = 13                 7 2     j = 11               8 4       j = 9             10 7         j = 7           9 8           j = 5         7 10             j = 3       6 9               j = 1     3 8                 j = -1   1 5                   j = -3   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, 41]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 41]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 5, 15, 6], X[2, 8, 3, 7], > X[18, 10, 19, 9], X[20, 12, 21, 11], X[16, 13, 17, 14], X[6, 15, 7, 16], > X[22, 18, 1, 17], X[12, 20, 13, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 41]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -3, 8, -7, 9, -5, 10, > -6, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 41]]
Out[5]=	DTCode[4, 8, 14, 2, 18, 20, 16, 6, 22, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 41]][t]
Out[6]=	2 11 27 2 3 -35 + -- - -- + -- + 27 t - 11 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 41]][z]
Out[7]=	2 4 6 1 + z + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 121], Knot[11, Alternating, 41], Knot[11, Alternating, 183], > Knot[11, Alternating, 198], Knot[11, Alternating, 331]}
In[9]:=	{KnotDet[Knot[11, Alternating, 41]], KnotSignature[Knot[11, Alternating, 41]]}
Out[9]=	{115, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 41]][q]
Out[10]=	-2 4 2 3 4 5 6 7 8 9 -8 - q + - + 13 q - 16 q + 19 q - 18 q + 15 q - 11 q + 6 q - 3 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 41], Knot[11, Alternating, 183]}
In[12]:=	A2Invariant[Knot[11, Alternating, 41]][q]
Out[12]=	-6 2 -2 2 4 6 8 12 14 16 18 -1 - q + -- - q + 4 q - 3 q + 3 q + q + 3 q - 3 q + 2 q - 2 q - 4 q 20 22 24 28 > 3 q + 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 41]][a, z]
Out[13]=	2 2 2 2 4 4 4 -8 3 2 -2 2 z 4 z 3 z 2 z 4 2 z 2 z 2 z a - -- + -- + a - z + -- - ---- + ---- + ---- - z - ---- + ---- + ---- + 6 4 8 6 4 2 6 4 2 a a a a a a a a a 6 6 z z > -- + -- 4 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 41]][a, z]
Out[14]=	2 2 2 -8 3 2 -2 3 z 8 z 7 z 3 z z 2 2 z 2 z 13 z a + -- + -- - a - --- - --- - --- - --- - - + 2 z + ---- - ---- - ----- - 6 4 9 7 5 3 a 10 8 6 a a a a a a a a a 2 2 3 3 3 3 3 4 10 z z 8 z 19 z 20 z 14 z 4 z 3 4 3 z > ----- + -- + ---- + ----- + ----- + ----- + ---- - a z - 6 z - ---- + 4 2 9 7 5 3 a 10 a a a a a a a 4 4 4 5 5 5 5 5 5 z 28 z 26 z 9 z 15 z 12 z 18 z 11 z 5 6 > ---- + ----- + ----- - ---- - ----- - ----- - ----- - ----- + a z + 4 z + 8 6 4 9 7 5 3 a a a a a a a a 6 6 6 6 6 7 7 7 7 8 z 9 z 27 z 29 z 8 z 3 z 6 z 4 z 7 z 4 z > --- - ---- - ----- - ----- - ---- + ---- - ---- + ---- + ---- + ---- + 10 8 6 4 2 9 5 3 a 8 a a a a a a a a a 8 8 8 9 9 9 10 10 9 z 12 z 7 z 3 z 7 z 4 z z z > ---- + ----- + ---- + ---- + ---- + ---- + --- + --- 6 4 2 7 5 3 6 4 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 41]], Vassiliev[3][Knot[11, Alternating, 41]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, Alternating, 41]][q, t]
Out[16]=	3 1 3 1 5 3 q 3 5 5 2 8 q + 6 q + ----- + ----- + ---- + --- + --- + 9 q t + 7 q t + 10 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 13 5 > 9 q t + 8 q t + 10 q t + 7 q t + 8 q t + 4 q t + 7 q t + 13 6 15 6 15 7 17 7 19 8 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a41

K11a40 K11a42