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

K11a6 K11a8

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 10 16 2 18 20 6 22 12 14

Alexander Polynomial:

- t-4 + 5t-3 - 12t-2 + 19t-1 - 21 + 19t - 12t2 + 5t3 - t4

Conway Polynomial:

1 - 2z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{10116, K11a33, K11a82, ...}

Determinant and Signature:

{95, -2}

Jones Polynomial:

- q-8 + 3q-7 - 6q-6 + 10q-5 - 13q-4 + 15q-3 - 15q-2 + 13q-1 - 9 + 6q - 3q2 + q3

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

{K11a325, ...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

3 + 5z2 + 4z4 + z6 - 6a2 - 15a2z2 - 14a2z4 - 6a2z6 - a2z8 + 6a4 + 13a4z2 + 9a4z4 + 2a4z6 - 2a6 - 3a6z2 - a6z4

Kauffman Polynomial:

a-2z2 - 3a-2z4 + a-2z6 - a-1z + 5a-1z3 - 9a-1z5 + 3a-1z7 + 3 - 10z2 + 18z4 - 17z6 + 5z8 - az + 2az3 + 5az5 - 10az7 + 4az9 + 6a2 - 31a2z2 + 55a2z4 - 37a2z6 + 8a2z8 + a2z10 - 6a3z3 + 21a3z5 - 19a3z7 + 7a3z9 + 6a4 - 27a4z2 + 41a4z4 - 26a4z6 + 7a4z8 + a4z10 - 2a5z3 + a5z5 - 2a5z7 + 3a5z9 + 2a6 - 4a6z2 + a6z4 - 4a6z6 + 4a6z8 + a7z - a7z3 - 5a7z5 + 4a7z7 + 3a8z2 - 6a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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

{0, -2}

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 117. 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                   4 1   j = 1                 5 2     j = -1               8 4       j = -3             8 6         j = -5           7 7           j = -7         6 8             j = -9       4 7               j = -11     2 6                 j = -13   1 4                   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, 7]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 7]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[16, 8, 17, 7], > X[2, 9, 3, 10], X[18, 11, 19, 12], X[20, 13, 21, 14], X[6, 16, 7, 15], > X[22, 17, 1, 18], X[12, 19, 13, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 7]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -11, 8, -4, 9, -6, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 7]]
Out[5]=	DTCode[4, 8, 10, 16, 2, 18, 20, 6, 22, 12, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 7]][t]
Out[6]=	-4 5 12 19 2 3 4 -21 - t + -- - -- + -- + 19 t - 12 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 7]][z]
Out[7]=	4 6 8 1 - 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 116], Knot[11, Alternating, 7], Knot[11, Alternating, 33], > Knot[11, Alternating, 82]}
In[9]:=	{KnotDet[Knot[11, Alternating, 7]], KnotSignature[Knot[11, Alternating, 7]]}
Out[9]=	{95, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 7]][q]
Out[10]=	-8 3 6 10 13 15 15 13 2 3 -9 - q + -- - -- + -- - -- + -- - -- + -- + 6 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, 7], Knot[11, Alternating, 325]}
In[12]:=	A2Invariant[Knot[11, Alternating, 7]][q]
Out[12]=	-24 -18 3 -14 2 -10 2 2 4 2 4 6 8 -q - q + --- - q + --- + q - -- + -- - -- + -- + 2 q - q + q 16 12 8 6 4 2 q q q q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 7]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 3 - 6 a + 6 a - 2 a + 5 z - 15 a z + 13 a z - 3 a z + 4 z - 2 4 4 4 6 4 6 2 6 4 6 2 8 > 14 a z + 9 a z - a z + z - 6 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 7]][a, z]
Out[14]=	2 2 4 6 z 7 9 2 z 2 2 3 + 6 a + 6 a + 2 a - - - a z + a z + a z - 10 z + -- - 31 a z - a 2 a 3 4 2 6 2 8 2 5 z 3 3 3 5 3 7 3 > 27 a z - 4 a z + 3 a z + ---- + 2 a z - 6 a z - 2 a z - a z - a 4 5 9 3 4 3 z 2 4 4 4 6 4 8 4 9 z > 2 a z + 18 z - ---- + 55 a z + 41 a z + a z - 6 a z - ---- + 2 a a 6 5 3 5 5 5 7 5 9 5 6 z 2 6 > 5 a z + 21 a z + a z - 5 a z + a z - 17 z + -- - 37 a z - 2 a 7 4 6 6 6 8 6 3 z 7 3 7 5 7 > 26 a z - 4 a z + 3 a z + ---- - 10 a z - 19 a z - 2 a z + a 7 7 8 2 8 4 8 6 8 9 3 9 5 9 > 4 a z + 5 z + 8 a z + 7 a z + 4 a z + 4 a z + 7 a z + 3 a z + 2 10 4 10 > a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 7]], Vassiliev[3][Knot[11, Alternating, 7]]}
Out[15]=	{0, -2}
In[16]:=	Kh[Knot[11, Alternating, 7]][q, t]
Out[16]=	6 8 1 2 1 4 2 6 4 7 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 6 8 7 7 8 4 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 5 q t + 2 q t + 4 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 K11a7

K11a6 K11a8