10 97

10_96

10_98

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10 97 Quick Notes

10 97 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X12,6,13,5 X8394 X2,9,3,10 X16,12,17,11 X10,18,11,17 X18,8,19,7 X20,14,1,13 X14,20,15,19 X6,16,7,15
Gauss code	1, -4, 3, -1, 2, -10, 7, -3, 4, -6, 5, -2, 8, -9, 10, -5, 6, -7, 9, -8
Dowker-Thistlethwaite code	4 8 12 18 2 16 20 6 10 14
Conway Notation	[.2.210.2]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-1][-11] Hyperbolic Volume 14.8527 A-Polynomial See Data:10 97/A-polynomial

[edit Notes for 10 97's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant -2

[edit Notes for 10 97's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-5t^{2}+22t-33+22t^{-1}-5t^{-2}$
Conway polynomial	$-5z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 87, 2 }
Jones polynomial	$q^{9}-4q^{8}+7q^{7}-11q^{6}+14q^{5}-14q^{4}+14q^{3}-11q^{2}+7q-3+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-3z^{4}a^{-4}-z^{4}a^{-6}+2z^{2}a^{-2}-4z^{2}a^{-4}+2z^{2}a^{-6}+z^{2}a^{-8}+z^{2}+2a^{-2}-2a^{-4}+2a^{-6}-a^{-8}$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-5}+2z^{9}a^{-7}+6z^{8}a^{-4}+11z^{8}a^{-6}+5z^{8}a^{-8}+8z^{7}a^{-3}+11z^{7}a^{-5}+7z^{7}a^{-7}+4z^{7}a^{-9}+6z^{6}a^{-2}-3z^{6}a^{-4}-21z^{6}a^{-6}-11z^{6}a^{-8}+z^{6}a^{-10}+3z^{5}a^{-1}-12z^{5}a^{-3}-32z^{5}a^{-5}-28z^{5}a^{-7}-11z^{5}a^{-9}-7z^{4}a^{-2}-9z^{4}a^{-4}+5z^{4}a^{-6}+4z^{4}a^{-8}-2z^{4}a^{-10}+z^{4}-2z^{3}a^{-1}+10z^{3}a^{-3}+24z^{3}a^{-5}+20z^{3}a^{-7}+8z^{3}a^{-9}+6z^{2}a^{-2}+10z^{2}a^{-4}+3z^{2}a^{-6}+z^{2}a^{-8}+z^{2}a^{-10}-z^{2}-2za^{-3}-6za^{-5}-4za^{-7}-2a^{-2}-2a^{-4}-2a^{-6}-a^{-8}$
The A2 invariant	$q^{4}-q^{2}-1+4q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+2q^{-12}-2q^{-14}+2q^{-16}+q^{-18}-2q^{-20}+3q^{-22}-2q^{-24}-2q^{-26}+q^{-28}$
The G2 invariant	$q^{18}-2q^{16}+4q^{14}-6q^{12}+6q^{10}-5q^{8}+10q^{4}-19q^{2}+31-39q^{-2}+37q^{-4}-22q^{-6}-8q^{-8}+53q^{-10}-94q^{-12}+123q^{-14}-126q^{-16}+82q^{-18}-q^{-20}-105q^{-22}+205q^{-24}-241q^{-26}+205q^{-28}-91q^{-30}-61q^{-32}+196q^{-34}-257q^{-36}+214q^{-38}-82q^{-40}-83q^{-42}+199q^{-44}-208q^{-46}+106q^{-48}+71q^{-50}-229q^{-52}+296q^{-54}-241q^{-56}+68q^{-58}+148q^{-60}-329q^{-62}+404q^{-64}-336q^{-66}+158q^{-68}+73q^{-70}-266q^{-72}+359q^{-74}-328q^{-76}+186q^{-78}+3q^{-80}-172q^{-82}+252q^{-84}-210q^{-86}+77q^{-88}+98q^{-90}-221q^{-92}+231q^{-94}-136q^{-96}-38q^{-98}+207q^{-100}-299q^{-102}+277q^{-104}-150q^{-106}-24q^{-108}+177q^{-110}-251q^{-112}+231q^{-114}-142q^{-116}+28q^{-118}+61q^{-120}-111q^{-122}+108q^{-124}-71q^{-126}+33q^{-128}+2q^{-130}-19q^{-132}+20q^{-134}-16q^{-136}+8q^{-138}-3q^{-140}+q^{-142}$

Further Quantum Invariants

Further quantum knot invariants for 10_97.

A1 Invariants.

Weight	Invariant
1	$q^{3}-2q+4q^{-1}-4q^{-3}+3q^{-5}+3q^{-11}-4q^{-13}+3q^{-15}-3q^{-17}+q^{-19}$
2	$q^{10}-2q^{8}+q^{6}+5q^{4}-11q^{2}+4+19q^{-2}-27q^{-4}-2q^{-6}+38q^{-8}-27q^{-10}-17q^{-12}+36q^{-14}-6q^{-16}-23q^{-18}+12q^{-20}+19q^{-22}-16q^{-24}-16q^{-26}+30q^{-28}-q^{-30}-37q^{-32}+25q^{-34}+18q^{-36}-37q^{-38}+8q^{-40}+24q^{-42}-18q^{-44}-5q^{-46}+12q^{-48}-2q^{-50}-3q^{-52}+q^{-54}$
3	$q^{21}-2q^{19}+q^{17}+2q^{15}-2q^{13}-5q^{11}+7q^{9}+13q^{7}-20q^{5}-29q^{3}+39q+61q^{-1}-51q^{-3}-119q^{-5}+54q^{-7}+189q^{-9}-27q^{-11}-248q^{-13}-37q^{-15}+284q^{-17}+116q^{-19}-271q^{-21}-184q^{-23}+206q^{-25}+236q^{-27}-116q^{-29}-245q^{-31}+13q^{-33}+227q^{-35}+78q^{-37}-186q^{-39}-153q^{-41}+137q^{-43}+215q^{-45}-89q^{-47}-257q^{-49}+27q^{-51}+285q^{-53}+37q^{-55}-287q^{-57}-119q^{-59}+266q^{-61}+183q^{-63}-202q^{-65}-235q^{-67}+117q^{-69}+250q^{-71}-26q^{-73}-215q^{-75}-48q^{-77}+155q^{-79}+86q^{-81}-85q^{-83}-86q^{-85}+28q^{-87}+60q^{-89}+4q^{-91}-34q^{-93}-9q^{-95}+11q^{-97}+7q^{-99}-2q^{-101}-3q^{-103}+q^{-105}$

A2 Invariants.

Weight	Invariant
1,0	$q^{4}-q^{2}-1+4q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+2q^{-12}-2q^{-14}+2q^{-16}+q^{-18}-2q^{-20}+3q^{-22}-2q^{-24}-2q^{-26}+q^{-28}$
2,0	$q^{12}-q^{10}-2q^{8}+3q^{6}+5q^{4}-4q^{2}-11+7q^{-2}+20q^{-4}-11q^{-6}-22q^{-8}+12q^{-10}+24q^{-12}-10q^{-14}-22q^{-16}+11q^{-18}+16q^{-20}-8q^{-22}-8q^{-24}+8q^{-26}+2q^{-30}+10q^{-32}-6q^{-34}-10q^{-36}+6q^{-38}+14q^{-40}-17q^{-42}-19q^{-44}+14q^{-46}+18q^{-48}-13q^{-50}-17q^{-52}+12q^{-54}+15q^{-56}-6q^{-58}-15q^{-60}+3q^{-62}+9q^{-64}+2q^{-66}-3q^{-68}-2q^{-70}+q^{-72}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{8}-2q^{6}+6q^{2}-7-4q^{-2}+19q^{-4}-11q^{-6}-13q^{-8}+30q^{-10}-11q^{-12}-18q^{-14}+29q^{-16}-7q^{-18}-15q^{-20}+13q^{-22}+3q^{-24}-6q^{-26}-7q^{-28}+11q^{-30}+9q^{-32}-23q^{-34}+9q^{-36}+18q^{-38}-29q^{-40}+5q^{-42}+18q^{-44}-22q^{-46}+7q^{-48}+11q^{-50}-12q^{-52}+4q^{-54}+2q^{-56}-3q^{-58}+q^{-60}$
1,0,0	$q^{5}-q^{3}-q^{-1}+4q^{-3}-2q^{-5}+3q^{-7}+2q^{-11}-2q^{-13}-2q^{-19}+2q^{-21}+3q^{-25}-2q^{-27}+3q^{-29}-2q^{-31}-q^{-33}-2q^{-35}+q^{-37}$

B2 Invariants.

Weight	Invariant
0,1	$q^{8}-2q^{6}+4q^{4}-8q^{2}+13-18q^{-2}+27q^{-4}-31q^{-6}+35q^{-8}-34q^{-10}+29q^{-12}-18q^{-14}+3q^{-16}+13q^{-18}-31q^{-20}+47q^{-22}-61q^{-24}+68q^{-26}-67q^{-28}+63q^{-30}-49q^{-32}+35q^{-34}-15q^{-36}-2q^{-38}+17q^{-40}-29q^{-42}+34q^{-44}-36q^{-46}+33q^{-48}-29q^{-50}+20q^{-52}-14q^{-54}+8q^{-56}-3q^{-58}+q^{-60}$
1,0	$q^{14}-2q^{10}-2q^{8}+2q^{6}+7q^{4}+2q^{2}-10-11q^{-2}+5q^{-4}+23q^{-6}+8q^{-8}-23q^{-10}-24q^{-12}+12q^{-14}+36q^{-16}+6q^{-18}-34q^{-20}-20q^{-22}+25q^{-24}+30q^{-26}-12q^{-28}-31q^{-30}+q^{-32}+29q^{-34}+7q^{-36}-25q^{-38}-12q^{-40}+20q^{-42}+16q^{-44}-15q^{-46}-19q^{-48}+14q^{-50}+24q^{-52}-8q^{-54}-32q^{-56}-2q^{-58}+34q^{-60}+17q^{-62}-30q^{-64}-34q^{-66}+14q^{-68}+37q^{-70}+4q^{-72}-31q^{-74}-18q^{-76}+19q^{-78}+23q^{-80}-5q^{-82}-16q^{-84}-4q^{-86}+9q^{-88}+5q^{-90}-3q^{-92}-3q^{-94}+q^{-98}$

G2 Invariants.

Weight

Invariant

1,0

$q^{18}-2q^{16}+4q^{14}-6q^{12}+6q^{10}-5q^{8}+10q^{4}-19q^{2}+31-39q^{-2}+37q^{-4}-22q^{-6}-8q^{-8}+53q^{-10}-94q^{-12}+123q^{-14}-126q^{-16}+82q^{-18}-q^{-20}-105q^{-22}+205q^{-24}-241q^{-26}+205q^{-28}-91q^{-30}-61q^{-32}+196q^{-34}-257q^{-36}+214q^{-38}-82q^{-40}-83q^{-42}+199q^{-44}-208q^{-46}+106q^{-48}+71q^{-50}-229q^{-52}+296q^{-54}-241q^{-56}+68q^{-58}+148q^{-60}-329q^{-62}+404q^{-64}-336q^{-66}+158q^{-68}+73q^{-70}-266q^{-72}+359q^{-74}-328q^{-76}+186q^{-78}+3q^{-80}-172q^{-82}+252q^{-84}-210q^{-86}+77q^{-88}+98q^{-90}-221q^{-92}+231q^{-94}-136q^{-96}-38q^{-98}+207q^{-100}-299q^{-102}+277q^{-104}-150q^{-106}-24q^{-108}+177q^{-110}-251q^{-112}+231q^{-114}-142q^{-116}+28q^{-118}+61q^{-120}-111q^{-122}+108q^{-124}-71q^{-126}+33q^{-128}+2q^{-130}-19q^{-132}+20q^{-134}-16q^{-136}+8q^{-138}-3q^{-140}+q^{-142}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 97"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-5t^{2}+22t-33+22t^{-1}-5t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-5z^{4}+2z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 87, 2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{9}-4q^{8}+7q^{7}-11q^{6}+14q^{5}-14q^{4}+14q^{3}-11q^{2}+7q-3+q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{4}a^{-2}-3z^{4}a^{-4}-z^{4}a^{-6}+2z^{2}a^{-2}-4z^{2}a^{-4}+2z^{2}a^{-6}+z^{2}a^{-8}+z^{2}+2a^{-2}-2a^{-4}+2a^{-6}-a^{-8}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$2z^{9}a^{-5}+2z^{9}a^{-7}+6z^{8}a^{-4}+11z^{8}a^{-6}+5z^{8}a^{-8}+8z^{7}a^{-3}+11z^{7}a^{-5}+7z^{7}a^{-7}+4z^{7}a^{-9}+6z^{6}a^{-2}-3z^{6}a^{-4}-21z^{6}a^{-6}-11z^{6}a^{-8}+z^{6}a^{-10}+3z^{5}a^{-1}-12z^{5}a^{-3}-32z^{5}a^{-5}-28z^{5}a^{-7}-11z^{5}a^{-9}-7z^{4}a^{-2}-9z^{4}a^{-4}+5z^{4}a^{-6}+4z^{4}a^{-8}-2z^{4}a^{-10}+z^{4}-2z^{3}a^{-1}+10z^{3}a^{-3}+24z^{3}a^{-5}+20z^{3}a^{-7}+8z^{3}a^{-9}+6z^{2}a^{-2}+10z^{2}a^{-4}+3z^{2}a^{-6}+z^{2}a^{-8}+z^{2}a^{-10}-z^{2}-2za^{-3}-6za^{-5}-4za^{-7}-2a^{-2}-2a^{-4}-2a^{-6}-a^{-8}$

Vassiliev invariants

V2 and V3:

(2, 4)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $8$ $32$ $32$ ${\frac {508}{3}}$ ${\frac {164}{3}}$ $256$ ${\frac {2432}{3}}$ ${\frac {512}{3}}$ $224$ ${\frac {256}{3}}$ $512$ ${\frac {4064}{3}}$ ${\frac {1312}{3}}$ ${\frac {55471}{15}}$ $-{\frac {2868}{5}}$ ${\frac {117724}{45}}$ ${\frac {593}{9}}$ ${\frac {6511}{15}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 10 97. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-2	-1	0	1	2	3	4	5	6	7	8	χ
19											1	1
17										3		-3
15									4	1		3
13								7	3			-4
11							7	4				3
9						7	7					0
7					7	7						0
5				4	7							3
3			3	7								-4
1		1	5									4
-1		2										-2
-3	1											1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 97]]
Out[2]=	10
In[3]:=	PD[Knot[10, 97]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 6, 13, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], X[16, 12, 17, 11], X[10, 18, 11, 17], X[18, 8, 19, 7], X[20, 14, 1, 13], X[14, 20, 15, 19], X[6, 16, 7, 15]]
In[4]:=	GaussCode[Knot[10, 97]]
Out[4]=	GaussCode[1, -4, 3, -1, 2, -10, 7, -3, 4, -6, 5, -2, 8, -9, 10, -5, 6, -7, 9, -8]
In[5]:=	BR[Knot[10, 97]]
Out[5]=	BR[5, {1, 1, 2, -1, 2, 1, -3, 2, -1, 2, 3, -4, 3, -4}]
In[6]:=	alex = Alexander[Knot[10, 97]][t]
Out[6]=	5 22 2 -33 - -- + -- + 22 t - 5 t 2 t t
In[7]:=	Conway[Knot[10, 97]][z]
Out[7]=	2 4 1 + 2 z - 5 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 97]}
In[9]:=	{KnotDet[Knot[10, 97]], KnotSignature[Knot[10, 97]]}
Out[9]=	{87, 2}
In[10]:=	J=Jones[Knot[10, 97]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 -3 + - + 7 q - 11 q + 14 q - 14 q + 14 q - 11 q + 7 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 97]}
In[12]:=	A2Invariant[Knot[10, 97]][q]
Out[12]=	-4 -2 2 4 6 8 10 12 14 -1 + q - q + 4 q - 2 q + q + 2 q - 2 q + 2 q - 2 q + 16 18 20 22 24 26 28 2 q + q - 2 q + 3 q - 2 q - 2 q + q
In[13]:=	Kauffman[Knot[10, 97]][a, z]
Out[13]=	2 2 2 2 -8 2 2 2 4 z 6 z 2 z 2 z z 3 z 10 z -a - -- - -- - -- - --- - --- - --- - z + --- + -- + ---- + ----- + 6 4 2 7 5 3 10 8 6 4 a a a a a a a a a a 2 3 3 3 3 3 4 4 6 z 8 z 20 z 24 z 10 z 2 z 4 2 z 4 z ---- + ---- + ----- + ----- + ----- - ---- + z - ---- + ---- + 2 9 7 5 3 a 10 8 a a a a a a a 4 4 4 5 5 5 5 5 6 5 z 9 z 7 z 11 z 28 z 32 z 12 z 3 z z ---- - ---- - ---- - ----- - ----- - ----- - ----- + ---- + --- - 6 4 2 9 7 5 3 a 10 a a a a a a a a 6 6 6 6 7 7 7 7 8 11 z 21 z 3 z 6 z 4 z 7 z 11 z 8 z 5 z ----- - ----- - ---- + ---- + ---- + ---- + ----- + ---- + ---- + 8 6 4 2 9 7 5 3 8 a a a a a a a a a 8 8 9 9 11 z 6 z 2 z 2 z ----- + ---- + ---- + ---- 6 4 7 5 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 97]], Vassiliev[3][Knot[10, 97]]}
Out[14]=	{0, 4}
In[15]:=	Kh[Knot[10, 97]][q, t]
Out[15]=	3 1 2 q 3 5 5 2 7 2 5 q + 3 q + ----- + --- + - + 7 q t + 4 q t + 7 q t + 7 q t + 3 2 q t t q t 7 3 9 3 9 4 11 4 11 5 13 5 7 q t + 7 q t + 7 q t + 7 q t + 4 q t + 7 q t + 13 6 15 6 15 7 17 7 19 8 3 q t + 4 q t + q t + 3 q t + q t