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

K11a2 K11a4

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X18,11,19,12 X6,14,7,13 X20,15,21,16 X22,17,1,18 X12,19,13,20 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 10 14 2 18 6 20 22 12 16

Alexander Polynomial:

- t-4 + 5t-3 - 13t-2 + 24t-1 - 29 + 24t - 13t2 + 5t3 - t4

Conway Polynomial:

1 + z2 - 3z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{115, -2}

Jones Polynomial:

- q-8 + 3q-7 - 7q-6 + 12q-5 - 16q-4 + 19q-3 - 18q-2 + 16q-1 - 12 + 7q - 3q2 + q3

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

{K11a51, K11a331, ...}

A2 (sl(3)) Invariant:

- q-24 - q-20 - 2q-18 + 4q-16 - q-14 + 3q-12 + 2q-10 - 2q-8 + 3q-6 - 5q-4 + 2q-2 - 1 - q2 + 3q4 - q6 + q8

HOMFLY-PT Polynomial:

3 + 6z2 + 4z4 + z6 - 7a2 - 17a2z2 - 15a2z4 - 6a2z6 - a2z8 + 8a4 + 15a4z2 + 9a4z4 + 2a4z6 - 3a6 - 3a6z2 - a6z4

Kauffman Polynomial:

2a-2z2 - 3a-2z4 + a-2z6 - 2a-1z + 6a-1z3 - 8a-1z5 + 3a-1z7 + 3 - 8z2 + 12z4 - 13z6 + 5z8 - 4az + 9az3 - 7az5 - 4az7 + 4az9 + 7a2 - 30a2z2 + 49a2z4 - 40a2z6 + 12a2z8 + a2z10 - 4a3z + 10a3z3 - 2a3z5 - 11a3z7 + 8a3z9 + 8a4 - 30a4z2 + 48a4z4 - 38a4z6 + 13a4z8 + a4z10 - 4a5z + 12a5z3 - 11a5z5 + a5z7 + 4a5z9 + 3a6 - 8a6z2 + 9a6z4 - 9a6z6 + 6a6z8 - a7z + 3a7z3 - 7a7z5 + 5a7z7 + 2a8z2 - 5a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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

{1, -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=-2 is the signature of 113. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     2   j = 3                   5 1   j = 1                 7 2     j = -1               9 5       j = -3             10 8         j = -5           9 8           j = -7         7 10             j = -9       5 9               j = -11     2 7                 j = -13   1 5                   j = -15   2                     j = -17 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, 3]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 3]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[14, 8, 15, 7], > X[2, 9, 3, 10], X[18, 11, 19, 12], X[6, 14, 7, 13], X[20, 15, 21, 16], > X[22, 17, 1, 18], X[12, 19, 13, 20], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 3]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -7, 4, -2, 5, -3, 6, -10, 7, -4, 8, -11, 9, -6, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 3]]
Out[5]=	DTCode[4, 8, 10, 14, 2, 18, 6, 20, 22, 12, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 3]][t]
Out[6]=	-4 5 13 24 2 3 4 -29 - t + -- - -- + -- + 24 t - 13 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 3]][z]
Out[7]=	2 4 6 8 1 + z - 3 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 3]}
In[9]:=	{KnotDet[Knot[11, Alternating, 3]], KnotSignature[Knot[11, Alternating, 3]]}
Out[9]=	{115, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 3]][q]
Out[10]=	-8 3 7 12 16 19 18 16 2 3 -12 - q + -- - -- + -- - -- + -- - -- + -- + 7 q - 3 q + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 3], Knot[11, Alternating, 51], > Knot[11, Alternating, 331]}
In[12]:=	A2Invariant[Knot[11, Alternating, 3]][q]
Out[12]=	-24 -20 2 4 -14 3 2 2 3 5 2 2 -1 - q - q - --- + --- - q + --- + --- - -- + -- - -- + -- - q + 18 16 12 10 8 6 4 2 q q q q q q q q 4 6 8 > 3 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 3]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 3 - 7 a + 8 a - 3 a + 6 z - 17 a z + 15 a z - 3 a z + 4 z - 2 4 4 4 6 4 6 2 6 4 6 2 8 > 15 a z + 9 a z - a z + z - 6 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 3]][a, z]
Out[14]=	2 4 6 2 z 3 5 7 9 2 3 + 7 a + 8 a + 3 a - --- - 4 a z - 4 a z - 4 a z - a z + a z - 8 z + a 2 3 2 z 2 2 4 2 6 2 8 2 6 z 3 3 3 > ---- - 30 a z - 30 a z - 8 a z + 2 a z + ---- + 9 a z + 10 a z + 2 a a 4 5 3 7 3 9 3 4 3 z 2 4 4 4 > 12 a z + 3 a z - 2 a z + 12 z - ---- + 49 a z + 48 a z + 2 a 5 6 4 8 4 8 z 5 3 5 5 5 7 5 9 5 > 9 a z - 5 a z - ---- - 7 a z - 2 a z - 11 a z - 7 a z + a z - a 6 7 6 z 2 6 4 6 6 6 8 6 3 z 7 > 13 z + -- - 40 a z - 38 a z - 9 a z + 3 a z + ---- - 4 a z - 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 11 a z + a z + 5 a z + 5 z + 12 a z + 13 a z + 6 a z + 9 3 9 5 9 2 10 4 10 > 4 a z + 8 a z + 4 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 3]], Vassiliev[3][Knot[11, Alternating, 3]]}
Out[15]=	{1, -4}
In[16]:=	Kh[Knot[11, Alternating, 3]][q, t]
Out[16]=	8 9 1 2 1 5 2 7 5 9 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 7 10 9 8 10 5 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 7 q t + 2 q t + 5 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a3

K11a2 K11a4