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

K11a43 K11a45

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 20 16 18 6 12 22 10

Alexander Polynomial:

t-4 - 5t-3 + 14t-2 - 24t-1 + 29 - 24t + 14t2 - 5t3 + t4

Conway Polynomial:

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

Other knots with the same Alexander/Conway Polynomial:

{K11a47, K11a109, ...}

Determinant and Signature:

{117, 0}

Jones Polynomial:

- q-5 + 3q-4 - 8q-3 + 13q-2 - 16q-1 + 20 - 18q + 16q2 - 12q3 + 6q4 - 3q5 + q6

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

{K11a47, ...}

A2 (sl(3)) Invariant:

- q-14 + q-12 - 4q-10 - q-4 + 8q-2 + 1 + 6q2 - q4 - 3q6 - 5q10 + q12 + q18

HOMFLY-PT Polynomial:

2a-4 + 3a-4z2 + a-4z4 - 9a-2 - 15a-2z2 - 9a-2z4 - 2a-2z6 + 13 + 22z2 + 16z4 + 6z6 + z8 - 5a2 - 7a2z2 - 4a2z4 - a2z6

Kauffman Polynomial:

2a-6z2 - 3a-6z4 + a-6z6 - 4a-5z + 9a-5z3 - 9a-5z5 + 3a-5z7 + 2a-4 - a-4z2 + 3a-4z4 - 8a-4z6 + 4a-4z8 - 15a-3z + 30a-3z3 - 22a-3z5 + 2a-3z7 + 3a-3z9 + 9a-2 - 19a-2z2 + 28a-2z4 - 28a-2z6 + 10a-2z8 + a-2z10 - 21a-1z + 42a-1z3 - 27a-1z5 - 2a-1z7 + 7a-1z9 + 13 - 28z2 + 37z4 - 33z6 + 13z8 + z10 - 15az + 31az3 - 25az5 + 5az7 + 4az9 + 5a2 - 11a2z2 + 11a2z4 - 11a2z6 + 7a2z8 - 4a3z + 8a3z3 - 10a3z5 + 6a3z7 + a4z2 - 4a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

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

{3, -2}

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=0 is the signature of 1144. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     2   j = 9                   4 1   j = 7                 8 2     j = 5               8 4       j = 3             10 8         j = 1           10 8           j = -1         7 11             j = -3       6 9               j = -5     2 7                 j = -7   1 6                   j = -9   2                     j = -11 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, 44]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 44]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 5, 15, 6], X[2, 8, 3, 7], > X[20, 10, 21, 9], X[16, 11, 17, 12], X[18, 13, 19, 14], X[6, 15, 7, 16], > X[12, 17, 13, 18], X[22, 20, 1, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 44]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -9, 7, -3, 8, -6, 9, -7, 10, > -5, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 44]]
Out[5]=	DTCode[4, 8, 14, 2, 20, 16, 18, 6, 12, 22, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 44]][t]
Out[6]=	-4 5 14 24 2 3 4 29 + t - -- + -- - -- - 24 t + 14 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 44]][z]
Out[7]=	2 4 6 8 1 + 3 z + 4 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 44], Knot[11, Alternating, 47], > Knot[11, Alternating, 109]}
In[9]:=	{KnotDet[Knot[11, Alternating, 44]], KnotSignature[Knot[11, Alternating, 44]]}
Out[9]=	{117, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 44]][q]
Out[10]=	-5 3 8 13 16 2 3 4 5 6 20 - q + -- - -- + -- - -- - 18 q + 16 q - 12 q + 6 q - 3 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 44], Knot[11, Alternating, 47]}
In[12]:=	A2Invariant[Knot[11, Alternating, 44]][q]
Out[12]=	-14 -12 4 -4 8 2 4 6 10 12 18 1 - q + q - --- - q + -- + 6 q - q - 3 q - 5 q + q + q 10 2 q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 44]][a, z]
Out[13]=	2 2 4 4 2 9 2 2 3 z 15 z 2 2 4 z 9 z 13 + -- - -- - 5 a + 22 z + ---- - ----- - 7 a z + 16 z + -- - ---- - 4 2 4 2 4 2 a a a a a a 6 2 4 6 2 z 2 6 8 > 4 a z + 6 z - ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 44]][a, z]
Out[14]=	2 9 2 4 z 15 z 21 z 3 5 2 13 + -- + -- + 5 a - --- - ---- - ---- - 15 a z - 4 a z + a z - 28 z + 4 2 5 3 a a a a a 2 2 2 3 3 3 2 z z 19 z 2 2 4 2 9 z 30 z 42 z 3 > ---- - -- - ----- - 11 a z + a z + ---- + ----- + ----- + 31 a z + 6 4 2 5 3 a a a a a a 4 4 4 3 3 5 3 4 3 z 3 z 28 z 2 4 4 4 > 8 a z - 2 a z + 37 z - ---- + ---- + ----- + 11 a z - 4 a z - 6 4 2 a a a 5 5 5 6 6 9 z 22 z 27 z 5 3 5 5 5 6 z 8 z > ---- - ----- - ----- - 25 a z - 10 a z + a z - 33 z + -- - ---- - 5 3 a 6 4 a a a a 6 7 7 7 28 z 2 6 4 6 3 z 2 z 2 z 7 3 7 > ----- - 11 a z + 3 a z + ---- + ---- - ---- + 5 a z + 6 a z + 2 5 3 a a a a 8 8 9 9 10 8 4 z 10 z 2 8 3 z 7 z 9 10 z > 13 z + ---- + ----- + 7 a z + ---- + ---- + 4 a z + z + --- 4 2 3 a 2 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 44]], Vassiliev[3][Knot[11, Alternating, 44]]}
Out[15]=	{3, -2}
In[16]:=	Kh[Knot[11, Alternating, 44]][q, t]
Out[16]=	11 1 2 1 6 2 7 6 9 -- + 10 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 7 3 3 2 5 2 5 3 7 3 7 4 > --- + 8 q t + 10 q t + 8 q t + 8 q t + 4 q t + 8 q t + 2 q t + q t 9 4 9 5 11 5 13 6 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a44

K11a43 K11a45