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

K11n64 K11n66

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,5,15,6 X2837 X18,9,19,10 X16,11,17,12 X13,20,14,21 X6,15,7,16 X10,17,11,18 X19,1,20,22 X21,12,22,13

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

3t-2 - 8t-1 + 11 - 8t + 3t2

Conway Polynomial:

1 + 4z2 + 3z4

Other knots with the same Alexander/Conway Polynomial:

{815, ...}

Determinant and Signature:

{33, 0}

Jones Polynomial:

- q-7 + 2q-6 - 4q-5 + 5q-4 - 5q-3 + 6q-2 - 4q-1 + 4 - 2q

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

{...}

A2 (sl(3)) Invariant:

- q-22 - 2q-16 - q-12 + q-10 + 3q-8 + 2q-6 + 3q-4 - 2q4

HOMFLY-PT Polynomial:

- 2 - 2z2 + 5a2 + 6a2z2 + 2a2z4 - a4 + a4z2 + a4z4 - a6 - a6z2

Kauffman Polynomial:

2a-1z - 2 + 2z2 + z4 + 4az - 3az3 - az5 + az7 - 5a2 + 5a2z2 + 2a2z4 - 6a2z6 + 2a2z8 + 2a3z + 2a3z3 - 5a3z5 - a3z7 + a3z9 - a4 - a4z2 + 12a4z4 - 15a4z6 + 4a4z8 - 4a5z + 13a5z3 - 9a5z5 - a5z7 + a5z9 + a6 - 4a6z2 + 11a6z4 - 9a6z6 + 2a6z8 - 4a7z + 8a7z3 - 5a7z5 + a7z7

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

{4, -5}

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 1165. 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 j = 3                 2 j = 1               2   j = -1             3 3   j = -3           3 1     j = -5         2 3       j = -7       3 3         j = -9     1 2           j = -11   1 3             j = -13   1               j = -15 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, 65]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 65]]
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, 9, 19, 10], X[16, 11, 17, 12], X[13, 20, 14, 21], X[6, 15, 7, 16], > X[10, 17, 11, 18], X[19, 1, 20, 22], X[21, 12, 22, 13]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 65]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -9, 6, 11, -7, -3, 8, -6, 9, -5, -10, > 7, -11, 10]
In[5]:=	DTCode[Knot[11, NonAlternating, 65]]
Out[5]=	DTCode[4, 8, 14, 2, 18, 16, -20, 6, 10, -22, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 65]][t]
Out[6]=	3 8 2 11 + -- - - - 8 t + 3 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 65]][z]
Out[7]=	2 4 1 + 4 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 15], Knot[11, NonAlternating, 65]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 65]], KnotSignature[Knot[11, NonAlternating, 65]]}
Out[9]=	{33, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 65]][q]
Out[10]=	-7 2 4 5 5 6 4 4 - q + -- - -- + -- - -- + -- - - - 2 q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 65]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 65]][q]
Out[12]=	-22 2 -12 -10 3 2 3 4 -q - --- - q + q + -- + -- + -- - 2 q 16 8 6 4 q q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 65]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 2 4 4 4 -2 + 5 a - a - a - 2 z + 6 a z + a z - a z + 2 a z + a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 65]][a, z]
Out[14]=	2 4 6 2 z 3 5 7 2 2 2 -2 - 5 a - a + a + --- + 4 a z + 2 a z - 4 a z - 4 a z + 2 z + 5 a z - a 4 2 6 2 3 3 3 5 3 7 3 4 2 4 > a z - 4 a z - 3 a z + 2 a z + 13 a z + 8 a z + z + 2 a z + 4 4 6 4 5 3 5 5 5 7 5 2 6 > 12 a z + 11 a z - a z - 5 a z - 9 a z - 5 a z - 6 a z - 4 6 6 6 7 3 7 5 7 7 7 2 8 4 8 > 15 a z - 9 a z + a z - a z - a z + a z + 2 a z + 4 a z + 6 8 3 9 5 9 > 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 65]], Vassiliev[3][Knot[11, NonAlternating, 65]]}
Out[15]=	{4, -5}
In[16]:=	Kh[Knot[11, NonAlternating, 65]][q, t]
Out[16]=	3 1 1 1 3 1 2 3 3 - + 2 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 2 3 3 1 3 3 > ----- + ----- + ----- + ---- + --- + 2 q t 5 3 5 2 3 2 3 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n65

K11n64 K11n66