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

K11n19 K11n21

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X7,17,8,16 X2,9,3,10 X11,18,12,19 X13,20,14,21 X15,22,16,1 X17,7,18,6 X19,14,20,15 X21,12,22,13

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 2t-2 + 6t-1 - 7 + 6t - 2t2

Conway Polynomial:

1 - 2z2 - 2z4

Other knots with the same Alexander/Conway Polynomial:

{86, K11n151, K11n152, ...}

Determinant and Signature:

{23, -2}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

q-16 + q-10 - q-8 - q-4 + 1 + q4 + q10

HOMFLY-PT Polynomial:

a-2 + a-2z2 - 2z2 - z4 - a2 - 2a2z2 - a2z4 + a4 + a4z2

Kauffman Polynomial:

- a-2 + 3a-2z2 - 4a-2z4 + a-2z6 - a-1z + 6a-1z3 - 8a-1z5 + 2a-1z7 + 6z4 - 8z6 + 2z8 - az + 4az3 - az5 - 3az7 + az9 + a2 - 10a2z2 + 22a2z4 - 15a2z6 + 3a2z8 - a3z - a3z3 + 7a3z5 - 5a3z7 + a3z9 + a4 - 7a4z2 + 12a4z4 - 6a4z6 + a4z8 - a5z + a5z3

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

{-2, 1}

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 1120. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                 1 j = 5               1   j = 3             2 1   j = 1         1 2 1     j = -1         3 2       j = -3       2 2         j = -5     1 2           j = -7   1 2             j = -9   1               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, NonAlternating, 20]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 20]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[7, 17, 8, 16], > X[2, 9, 3, 10], X[11, 18, 12, 19], X[13, 20, 14, 21], X[15, 22, 16, 1], > X[17, 7, 18, 6], X[19, 14, 20, 15], X[21, 12, 22, 13]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 20]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, 9, -4, -2, 5, -3, -6, 11, -7, 10, -8, 4, -9, 6, -10, > 7, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 20]]
Out[5]=	DTCode[4, 8, 10, -16, 2, -18, -20, -22, -6, -14, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 20]][t]
Out[6]=	2 6 2 -7 - -- + - + 6 t - 2 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 20]][z]
Out[7]=	2 4 1 - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 6], Knot[11, NonAlternating, 20], Knot[11, NonAlternating, 151], > Knot[11, NonAlternating, 152]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 20]], KnotSignature[Knot[11, NonAlternating, 20]]}
Out[9]=	{23, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 20]][q]
Out[10]=	-5 2 3 4 4 2 3 -3 + q - -- + -- - -- + - + 3 q - 2 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, NonAlternating, 20]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 20]][q]
Out[12]=	-16 -10 -8 -4 4 10 1 + q + q - q - q + q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 20]][a, z]
Out[13]=	2 -2 2 4 2 z 2 2 4 2 4 2 4 a - a + a - 2 z + -- - 2 a z + a z - z - a z 2 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 20]][a, z]
Out[14]=	2 3 -2 2 4 z 3 5 3 z 2 2 4 2 6 z -a + a + a - - - a z - a z - a z + ---- - 10 a z - 7 a z + ---- + a 2 a a 4 5 3 3 3 5 3 4 4 z 2 4 4 4 8 z 5 > 4 a z - a z + a z + 6 z - ---- + 22 a z + 12 a z - ---- - a z + 2 a a 6 7 3 5 6 z 2 6 4 6 2 z 7 3 7 8 > 7 a z - 8 z + -- - 15 a z - 6 a z + ---- - 3 a z - 5 a z + 2 z + 2 a a 2 8 4 8 9 3 9 > 3 a z + a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 20]], Vassiliev[3][Knot[11, NonAlternating, 20]]}
Out[15]=	{-2, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 20]][q, t]
Out[16]=	2 3 1 1 1 2 1 2 2 2 t -- + - + q + ------ + ----- + ----- + ----- + ----- + ---- + ---- + --- + 3 q 11 4 9 3 7 3 7 2 5 2 5 3 q q q t q t q t q t q t q t q t 2 3 2 3 3 5 3 7 4 > 2 q t + q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n20

K11n19 K11n21