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

K11n5 K11n7

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X11,18,12,19 X6,14,7,13 X15,20,16,21 X17,12,18,13 X19,22,20,1 X21,16,22,17

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 3t-2 - 3t-1 + 3 - 3t + 3t2 - t3

Conway Polynomial:

1 - 3z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{82, ...}

Determinant and Signature:

{17, 0}

Jones Polynomial:

- q-7 + 2q-6 - 3q-5 + 4q-4 - 3q-3 + 3q-2 - 2q-1 + 1 - q2 + q3

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

{...}

A2 (sl(3)) Invariant:

- q-22 - q-20 + 2q-14 + q-12 + 2q-10 + q-8 - q-6 - 2q-2 + q6 + q10

HOMFLY-PT Polynomial:

a-2 + a-2z2 - 3a2 - 7a2z2 - 5a2z4 - a2z6 + 5a4 + 7a4z2 + 2a4z4 - 2a6 - a6z2

Kauffman Polynomial:

- a-2 + 5a-2z2 - 5a-2z4 + a-2z6 - 3a-1z + 8a-1z3 - 6a-1z5 + a-1z7 + 4z2 - 3z4 - 6az + 13az3 - 8az5 + az7 + 3a2 - 12a2z2 + 19a2z4 - 12a2z6 + 2a2z8 - 6a3z + 13a3z3 - 5a3z5 - 3a3z7 + a3z9 + 5a4 - 19a4z2 + 31a4z4 - 21a4z6 + 4a4z8 - 6a5z + 15a5z3 - 8a5z5 - 2a5z7 + a5z9 + 2a6 - 8a6z2 + 14a6z4 - 10a6z6 + 2a6z8 - 3a7z + 7a7z3 - 5a7z5 + a7z7

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

{0, -3}

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 116. 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                         j = 3                 1 1 1   j = 1               2 1       j = -1             2 2 1       j = -3           2 2 1         j = -5         2 2             j = -7       2 2 1             j = -9     1 2                 j = -11   1 2                   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, 6]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 6]]
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[11, 18, 12, 19], X[6, 14, 7, 13], X[15, 20, 16, 21], > X[17, 12, 18, 13], X[19, 22, 20, 1], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 6]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -7, 4, -2, 5, -3, -6, 9, 7, -4, -8, 11, -9, 6, -10, > 8, -11, 10]
In[5]:=	DTCode[Knot[11, NonAlternating, 6]]
Out[5]=	DTCode[4, 8, 10, 14, 2, -18, 6, -20, -12, -22, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 6]][t]
Out[6]=	-3 3 3 2 3 3 - t + -- - - - 3 t + 3 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 6]][z]
Out[7]=	4 6 1 - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 2], Knot[11, NonAlternating, 6]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 6]], KnotSignature[Knot[11, NonAlternating, 6]]}
Out[9]=	{17, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 6]][q]
Out[10]=	-7 2 3 4 3 3 2 2 3 1 - q + -- - -- + -- - -- + -- - - - q + 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, 6]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 6]][q]
Out[12]=	-22 -20 2 -12 2 -8 -6 2 6 10 -q - q + --- + q + --- + q - q - -- + q + q 14 10 2 q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 6]][a, z]
Out[13]=	2 -2 2 4 6 z 2 2 4 2 6 2 2 4 4 4 a - 3 a + 5 a - 2 a + -- - 7 a z + 7 a z - a z - 5 a z + 2 a z - 2 a 2 6 > a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 6]][a, z]
Out[14]=	-2 2 4 6 3 z 3 5 7 2 -a + 3 a + 5 a + 2 a - --- - 6 a z - 6 a z - 6 a z - 3 a z + 4 z + a 2 3 5 z 2 2 4 2 6 2 8 z 3 3 3 > ---- - 12 a z - 19 a z - 8 a z + ---- + 13 a z + 13 a z + 2 a a 4 5 5 3 7 3 4 5 z 2 4 4 4 6 4 6 z > 15 a z + 7 a z - 3 z - ---- + 19 a z + 31 a z + 14 a z - ---- - 2 a a 6 5 3 5 5 5 7 5 z 2 6 4 6 > 8 a z - 5 a z - 8 a z - 5 a z + -- - 12 a z - 21 a z - 2 a 7 6 6 z 7 3 7 5 7 7 7 2 8 4 8 > 10 a z + -- + a z - 3 a z - 2 a z + a z + 2 a z + 4 a z + a 6 8 3 9 5 9 > 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 6]], Vassiliev[3][Knot[11, NonAlternating, 6]]}
Out[15]=	{0, -3}
In[16]:=	Kh[Knot[11, NonAlternating, 6]][q, t]
Out[16]=	-3 2 1 1 1 2 1 2 2 q + - + 2 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 q t q t q t q t q t q t q t 2 2 1 2 2 2 2 t 3 > ----- + ----- + ----- + ----- + ----- + ---- + --- + - + q t + q t + 7 3 5 3 7 2 5 2 3 2 3 q t q q t q t q t q t q t q t 3 2 3 3 7 4 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n6

K11n5 K11n7