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

K11n40 K11n42

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,13,6,12 X2837 X9,18,10,19 X11,17,12,16 X13,7,14,6 X15,21,16,20 X17,22,18,1 X19,15,20,14 X21,10,22,11

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-4 - 4t-3 + 8t-2 - 9t-1 + 9 - 9t + 8t2 - 4t3 + t4

Conway Polynomial:

1 + 3z2 + 4z4 + 4z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{K11n47, ...}

Determinant and Signature:

{53, 4}

Jones Polynomial:

- q-1 + 3 - 5q + 8q2 - 8q3 + 9q4 - 8q5 + 6q6 - 4q7 + q8

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

{K11n47, ...}

A2 (sl(3)) Invariant:

- q-2 + 1 - q2 + q4 + q6 + q8 + 4q10 - q12 + 3q14 - 2q16 - q18 - q20 - 2q22 + q24 - q26 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 5a-6 - 8a-6z2 - 5a-6z4 - a-6z6 + 6a-4 + 14a-4z2 + 13a-4z4 + 6a-4z6 + a-4z8 - a-2 - 4a-2z2 - 4a-2z4 - a-2z6

Kauffman Polynomial:

a-10z2 - 2a-9z + 4a-9z3 + a-8 - a-8z4 + 2a-8z6 - 10a-7z + 24a-7z3 - 20a-7z5 + 6a-7z7 + 5a-6 - 15a-6z2 + 24a-6z4 - 21a-6z6 + 6a-6z8 - 13a-5z + 35a-5z3 - 28a-5z5 + 2a-5z7 + 2a-5z9 + 6a-4 - 22a-4z2 + 42a-4z4 - 36a-4z6 + 9a-4z8 - 7a-3z + 20a-3z3 - 12a-3z5 - 3a-3z7 + 2a-3z9 + a-2 - 8a-2z2 + 17a-2z4 - 13a-2z6 + 3a-2z8 - 2a-1z + 5a-1z3 - 4a-1z5 + a-1z7

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

{3, 4}

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 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 j = 17                   1 j = 15                 3   j = 13               3 1   j = 11             5 3     j = 9           4 3       j = 7         4 5         j = 5       4 4           j = 3     2 5             j = 1   1 3               j = -1   2                 j = -3 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, NonAlternating, 41]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 41]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 13, 6, 12], X[2, 8, 3, 7], > X[9, 18, 10, 19], X[11, 17, 12, 16], X[13, 7, 14, 6], X[15, 21, 16, 20], > X[17, 22, 18, 1], X[19, 15, 20, 14], X[21, 10, 22, 11]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 41]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, -5, 11, -6, 3, -7, 10, -8, 6, -9, 5, -10, > 8, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 41]]
Out[5]=	DTCode[4, 8, -12, 2, -18, -16, -6, -20, -22, -14, -10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 41]][t]
Out[6]=	-4 4 8 9 2 3 4 9 + t - -- + -- - - - 9 t + 8 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 41]][z]
Out[7]=	2 4 6 8 1 + 3 z + 4 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 41], Knot[11, NonAlternating, 47]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 41]], KnotSignature[Knot[11, NonAlternating, 41]]}
Out[9]=	{53, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 41]][q]
Out[10]=	1 2 3 4 5 6 7 8 3 - - - 5 q + 8 q - 8 q + 9 q - 8 q + 6 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 41], Knot[11, NonAlternating, 47]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 41]][q]
Out[12]=	-2 2 4 6 8 10 12 14 16 18 20 22 1 - q - q + q + q + q + 4 q - q + 3 q - 2 q - q - q - 2 q + 24 26 28 > q - q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 41]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -8 5 6 -2 z 8 z 14 z 4 z 5 z 13 z 4 z z a - -- + -- - a + -- - ---- + ----- - ---- - ---- + ----- - ---- - -- + 6 4 8 6 4 2 6 4 2 6 a a a a a a a a a a 6 6 8 6 z z z > ---- - -- + -- 4 2 4 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 41]][a, z]
Out[14]=	2 2 2 -8 5 6 -2 2 z 10 z 13 z 7 z 2 z z 15 z 22 z a + -- + -- + a - --- - ---- - ---- - --- - --- + --- - ----- - ----- - 6 4 9 7 5 3 a 10 6 4 a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 8 z 4 z 24 z 35 z 20 z 5 z z 24 z 42 z 17 z > ---- + ---- + ----- + ----- + ----- + ---- - -- + ----- + ----- + ----- - 2 9 7 5 3 a 8 6 4 2 a a a a a a a a a 5 5 5 5 6 6 6 6 7 7 20 z 28 z 12 z 4 z 2 z 21 z 36 z 13 z 6 z 2 z > ----- - ----- - ----- - ---- + ---- - ----- - ----- - ----- + ---- + ---- - 7 5 3 a 8 6 4 2 7 5 a a a a a a a a a 7 7 8 8 8 9 9 3 z z 6 z 9 z 3 z 2 z 2 z > ---- + -- + ---- + ---- + ---- + ---- + ---- 3 a 6 4 2 5 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 41]], Vassiliev[3][Knot[11, NonAlternating, 41]]}
Out[15]=	{3, 4}
In[16]:=	Kh[Knot[11, NonAlternating, 41]][q, t]
Out[16]=	3 3 5 1 2 q 3 q 2 q 5 7 7 2 5 q + 4 q + ----- + ---- + -- + --- + ---- + 4 q t + 4 q t + 5 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 15 5 > 4 q t + 3 q t + 5 q t + 3 q t + 3 q t + q t + 3 q t + 17 6 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n41

K11n40 K11n42