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

K11a14 K11a16

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 12 2 14 18 6 20 22 10 16

Alexander Polynomial:

- t-4 + 5t-3 - 13t-2 + 22t-1 - 25 + 22t - 13t2 + 5t3 - t4

Conway Polynomial:

1 - z2 - 3z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{107, -2}

Jones Polynomial:

q-7 - 3q-6 + 7q-5 - 12q-4 + 15q-3 - 17q-2 + 17q-1 - 14 + 11q - 6q2 + 3q3 - q4

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

{...}

A2 (sl(3)) Invariant:

q-20 - q-18 + 3q-16 - q-14 - q-12 + q-10 - 5q-8 + 2q-6 - 3q-4 + 2q-2 + 3 + 4q4 - q6 - q12

HOMFLY-PT Polynomial:

- 2a-2 - 3a-2z2 - a-2z4 + 8 + 14z2 + 9z4 + 2z6 - 8a2 - 18a2z2 - 15a2z4 - 6a2z6 - a2z8 + 3a4 + 6a4z2 + 4a4z4 + a4z6

Kauffman Polynomial:

- 2a-3z + 5a-3z3 - 4a-3z5 + a-3z7 + 2a-2 - 9a-2z2 + 16a-2z4 - 12a-2z6 + 3a-2z8 - 6a-1z + 15a-1z3 - 6a-1z5 - 6a-1z7 + 3a-1z9 + 8 - 31z2 + 54z4 - 42z6 + 9z8 + z10 - 10az + 23az3 - 9az5 - 15az7 + 8az9 + 8a2 - 33a2z2 + 58a2z4 - 52a2z6 + 15a2z8 + a2z10 - 11a3z + 29a3z3 - 25a3z5 + a3z7 + 5a3z9 + 3a4 - 7a4z2 + 13a4z4 - 16a4z6 + 9a4z8 - 5a5z + 14a5z3 - 15a5z5 + 9a5z7 + 3a6z2 - 6a6z4 + 6a6z6 - 2a7z3 + 3a7z5 - a8z2 + a8z4

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

{-1, 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=-2 is the signature of 1115. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 9                       1 j = 7                     2   j = 5                   4 1   j = 3                 7 2     j = 1               7 4       j = -1             10 7         j = -3           8 8           j = -5         7 9             j = -7       5 8               j = -9     2 7                 j = -11   1 5                   j = -13   2                     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, Alternating, 15]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 15]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[14, 9, 15, 10], X[18, 12, 19, 11], X[6, 13, 7, 14], X[20, 15, 21, 16], > X[22, 17, 1, 18], X[10, 20, 11, 19], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 15]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -5, 8, -11, 9, -6, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 15]]
Out[5]=	DTCode[4, 8, 12, 2, 14, 18, 6, 20, 22, 10, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 15]][t]
Out[6]=	-4 5 13 22 2 3 4 -25 - t + -- - -- + -- + 22 t - 13 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 15]][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, 15]}
In[9]:=	{KnotDet[Knot[11, Alternating, 15]], KnotSignature[Knot[11, Alternating, 15]]}
Out[9]=	{107, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 15]][q]
Out[10]=	-7 3 7 12 15 17 17 2 3 4 -14 + q - -- + -- - -- + -- - -- + -- + 11 q - 6 q + 3 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, Alternating, 15]}
In[12]:=	A2Invariant[Knot[11, Alternating, 15]][q]
Out[12]=	-20 -18 3 -14 -12 -10 5 2 3 2 4 6 12 3 + q - q + --- - q - q + q - -- + -- - -- + -- + 4 q - q - q 16 8 6 4 2 q q q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 15]][a, z]
Out[13]=	2 4 2 2 4 2 3 z 2 2 4 2 4 z 8 - -- - 8 a + 3 a + 14 z - ---- - 18 a z + 6 a z + 9 z - -- - 2 2 2 a a a 2 4 4 4 6 2 6 4 6 2 8 > 15 a z + 4 a z + 2 z - 6 a z + a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 15]][a, z]
Out[14]=	2 2 2 4 2 z 6 z 3 5 2 9 z 8 + -- + 8 a + 3 a - --- - --- - 10 a z - 11 a z - 5 a z - 31 z - ---- - 2 3 a 2 a a a 3 3 2 2 4 2 6 2 8 2 5 z 15 z 3 3 3 > 33 a z - 7 a z + 3 a z - a z + ---- + ----- + 23 a z + 29 a z + 3 a a 4 5 3 7 3 4 16 z 2 4 4 4 6 4 > 14 a z - 2 a z + 54 z + ----- + 58 a z + 13 a z - 6 a z + 2 a 5 5 8 4 4 z 6 z 5 3 5 5 5 7 5 6 > a z - ---- - ---- - 9 a z - 25 a z - 15 a z + 3 a z - 42 z - 3 a a 6 7 7 12 z 2 6 4 6 6 6 z 6 z 7 3 7 > ----- - 52 a z - 16 a z + 6 a z + -- - ---- - 15 a z + a z + 2 3 a a a 8 9 5 7 8 3 z 2 8 4 8 3 z 9 3 9 > 9 a z + 9 z + ---- + 15 a z + 9 a z + ---- + 8 a z + 5 a z + 2 a a 10 2 10 > z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 15]], Vassiliev[3][Knot[11, Alternating, 15]]}
Out[15]=	{-1, 3}
In[16]:=	Kh[Knot[11, Alternating, 15]][q, t]
Out[16]=	8 10 1 2 1 5 2 7 5 8 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 7 9 8 7 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 7 q t + 4 q t + 7 q t + 2 q t + 4 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a15

K11a14 K11a16