9 12

9_11

9_13

Visit 9 12's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 12's page at Knotilus! Visit 9 12's page at the original Knot Atlas!

9 12 Quick Notes

9 12 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X5,16,6,17 X11,1,12,18 X17,13,18,12 X7,14,8,15 X13,8,14,9 X15,6,16,7 X9,2,10,3
Gauss code	-1, 9, -2, 1, -3, 8, -6, 7, -9, 2, -4, 5, -7, 6, -8, 3, -5, 4
Dowker-Thistlethwaite code	4 10 16 14 2 18 8 6 12
Conway Notation	[4212]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-10][-1] Hyperbolic Volume 8.83664 A-Polynomial See Data:9 12/A-polynomial

[edit Notes for 9 12'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 9 12's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+9t-13+9t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 35, -2 }
Jones polynomial	$q-2+4q^{-1}-5q^{-2}+6q^{-3}-6q^{-4}+5q^{-5}-3q^{-6}+2q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{8}+2z^{2}a^{6}+2a^{6}-z^{4}a^{4}-z^{2}a^{4}-a^{4}-z^{4}a^{2}-z^{2}a^{2}+z^{2}+1$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-3z^{3}a^{9}+za^{9}+2z^{6}a^{8}-6z^{4}a^{8}+4z^{2}a^{8}-a^{8}+2z^{7}a^{7}-5z^{5}a^{7}+3z^{3}a^{7}-za^{7}+z^{8}a^{6}-5z^{4}a^{6}+7z^{2}a^{6}-2a^{6}+4z^{7}a^{5}-11z^{5}a^{5}+13z^{3}a^{5}-4za^{5}+z^{8}a^{4}-z^{4}a^{4}+3z^{2}a^{4}-a^{4}+2z^{7}a^{3}-3z^{5}a^{3}+4z^{3}a^{3}-2za^{3}+2z^{6}a^{2}-z^{4}a^{2}-2z^{2}a^{2}+2z^{5}a-3z^{3}a+z^{4}-2z^{2}+1$
The A2 invariant	$-q^{26}-q^{24}+q^{22}+q^{18}+2q^{16}-q^{14}-q^{10}+q^{6}-q^{4}+2q^{2}+q^{-4}$
The G2 invariant	$q^{128}-q^{126}+2q^{124}-3q^{122}+2q^{120}-2q^{118}-2q^{116}+7q^{114}-10q^{112}+10q^{110}-10q^{108}+4q^{106}+4q^{104}-15q^{102}+20q^{100}-20q^{98}+14q^{96}-q^{94}-12q^{92}+19q^{90}-18q^{88}+15q^{86}-3q^{84}-10q^{82}+14q^{80}-11q^{78}+2q^{76}+10q^{74}-16q^{72}+18q^{70}-8q^{68}-4q^{66}+17q^{64}-26q^{62}+29q^{60}-21q^{58}+6q^{56}+11q^{54}-23q^{52}+30q^{50}-25q^{48}+13q^{46}-13q^{42}+16q^{40}-14q^{38}+3q^{36}+8q^{34}-13q^{32}+10q^{30}-2q^{28}-9q^{26}+17q^{24}-19q^{22}+15q^{20}-6q^{18}-5q^{16}+14q^{14}-16q^{12}+17q^{10}-10q^{8}+5q^{6}+q^{4}-7q^{2}+9-8q^{-2}+7q^{-4}-3q^{-6}+q^{-8}+2q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 9_12.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+q^{15}-q^{13}+2q^{11}-q^{9}+q^{5}-q^{3}+2q-q^{-1}+q^{-3}$
2	$q^{48}-q^{46}-q^{44}+3q^{42}-2q^{40}-4q^{38}+5q^{36}+q^{34}-6q^{32}+5q^{30}+3q^{28}-7q^{26}+2q^{24}+3q^{22}-3q^{20}-2q^{18}+2q^{16}+4q^{14}-5q^{12}+7q^{8}-5q^{6}-2q^{4}+7q^{2}-2-2q^{-2}+3q^{-4}-q^{-6}-q^{-8}+q^{-10}$
3	$-q^{93}+q^{91}+q^{89}-q^{87}-2q^{85}+2q^{83}+5q^{81}-2q^{79}-8q^{77}+10q^{73}+3q^{71}-12q^{69}-10q^{67}+12q^{65}+15q^{63}-6q^{61}-19q^{59}+3q^{57}+21q^{55}+2q^{53}-21q^{51}-7q^{49}+19q^{47}+7q^{45}-14q^{43}-9q^{41}+9q^{39}+10q^{37}-5q^{35}-10q^{33}-3q^{31}+11q^{29}+10q^{27}-10q^{25}-16q^{23}+9q^{21}+20q^{19}-5q^{17}-21q^{15}+q^{13}+20q^{11}+4q^{9}-15q^{7}-5q^{5}+11q^{3}+5q-5q^{-1}-4q^{-3}+3q^{-5}+2q^{-7}-2q^{-9}+2q^{-13}-q^{-17}-q^{-19}+q^{-21}$
4	$q^{152}-q^{150}-q^{148}+q^{146}+2q^{142}-4q^{140}-3q^{138}+4q^{136}+3q^{134}+8q^{132}-8q^{130}-13q^{128}+2q^{126}+8q^{124}+23q^{122}-4q^{120}-25q^{118}-17q^{116}-4q^{114}+40q^{112}+23q^{110}-12q^{108}-36q^{106}-41q^{104}+28q^{102}+50q^{100}+33q^{98}-24q^{96}-77q^{94}-14q^{92}+44q^{90}+69q^{88}+12q^{86}-80q^{84}-49q^{82}+16q^{80}+74q^{78}+38q^{76}-56q^{74}-51q^{72}-5q^{70}+52q^{68}+39q^{66}-25q^{64}-39q^{62}-15q^{60}+29q^{58}+32q^{56}+3q^{54}-30q^{52}-29q^{50}+31q^{46}+43q^{44}-17q^{42}-50q^{40}-36q^{38}+24q^{36}+79q^{34}+13q^{32}-50q^{30}-68q^{28}-5q^{26}+84q^{24}+42q^{22}-18q^{20}-67q^{18}-38q^{16}+48q^{14}+42q^{12}+18q^{10}-33q^{8}-40q^{6}+10q^{4}+14q^{2}+24-q^{-2}-18q^{-4}-2q^{-6}-5q^{-8}+10q^{-10}+4q^{-12}-3q^{-14}+3q^{-16}-6q^{-18}+4q^{-26}-q^{-28}-q^{-32}-q^{-34}+q^{-36}$
5	$-q^{225}+q^{223}+q^{221}-q^{219}+2q^{211}+2q^{209}-4q^{207}-6q^{205}-q^{203}+4q^{201}+9q^{199}+8q^{197}-4q^{195}-19q^{193}-17q^{191}+3q^{189}+23q^{187}+33q^{185}+13q^{183}-24q^{181}-50q^{179}-36q^{177}+12q^{175}+56q^{173}+66q^{171}+23q^{169}-47q^{167}-91q^{165}-71q^{163}+6q^{161}+91q^{159}+119q^{157}+58q^{155}-56q^{153}-149q^{151}-138q^{149}-4q^{147}+147q^{145}+202q^{143}+99q^{141}-106q^{139}-247q^{137}-186q^{135}+40q^{133}+245q^{131}+256q^{129}+44q^{127}-220q^{125}-299q^{123}-113q^{121}+170q^{119}+300q^{117}+164q^{115}-109q^{113}-278q^{111}-188q^{109}+65q^{107}+231q^{105}+181q^{103}-19q^{101}-182q^{99}-164q^{97}-3q^{95}+138q^{93}+134q^{91}+17q^{89}-96q^{87}-114q^{85}-35q^{83}+68q^{81}+106q^{79}+51q^{77}-34q^{75}-103q^{73}-93q^{71}+q^{69}+108q^{67}+140q^{65}+48q^{63}-107q^{61}-193q^{59}-114q^{57}+89q^{55}+238q^{53}+187q^{51}-47q^{49}-263q^{47}-256q^{45}-19q^{43}+249q^{41}+307q^{39}+93q^{37}-201q^{35}-316q^{33}-160q^{31}+122q^{29}+287q^{27}+207q^{25}-35q^{23}-224q^{21}-210q^{19}-36q^{17}+139q^{15}+184q^{13}+84q^{11}-65q^{9}-135q^{7}-90q^{5}+7q^{3}+80q+81q^{-1}+26q^{-3}-39q^{-5}-57q^{-7}-32q^{-9}+8q^{-11}+33q^{-13}+28q^{-15}+7q^{-17}-15q^{-19}-20q^{-21}-8q^{-23}+3q^{-25}+9q^{-27}+9q^{-29}+4q^{-31}-5q^{-33}-6q^{-35}-2q^{-37}-2q^{-39}+2q^{-41}+4q^{-43}+q^{-45}-q^{-47}-q^{-51}-q^{-53}+q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{26}-q^{24}+q^{22}+q^{18}+2q^{16}-q^{14}-q^{10}+q^{6}-q^{4}+2q^{2}+q^{-4}$
1,1	$q^{68}-2q^{66}+4q^{64}-8q^{62}+17q^{60}-24q^{58}+32q^{56}-48q^{54}+57q^{52}-64q^{50}+62q^{48}-58q^{46}+46q^{44}-18q^{42}-6q^{40}+38q^{38}-65q^{36}+90q^{34}-112q^{32}+116q^{30}-122q^{28}+110q^{26}-90q^{24}+68q^{22}-37q^{20}+10q^{18}+20q^{16}-36q^{14}+47q^{12}-54q^{10}+56q^{8}-48q^{6}+42q^{4}-36q^{2}+30-20q^{-2}+15q^{-4}-10q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$
2,0	$q^{66}+q^{64}-2q^{60}-q^{58}+q^{56}-q^{54}-3q^{52}-q^{50}+4q^{48}+3q^{46}-2q^{44}+4q^{40}-5q^{36}-3q^{34}+2q^{32}-2q^{28}+2q^{26}+q^{24}-q^{22}+3q^{20}+2q^{18}-3q^{16}-q^{14}+4q^{12}+q^{10}-4q^{8}+6q^{4}-3+q^{-2}+2q^{-4}-q^{-8}+q^{-12}$

A3 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{54}+q^{52}-2q^{50}+3q^{48}-5q^{46}+5q^{44}-6q^{42}+5q^{40}-4q^{38}+4q^{36}-q^{32}+6q^{30}-7q^{28}+10q^{26}-11q^{24}+11q^{22}-11q^{20}+7q^{18}-6q^{16}+2q^{14}-2q^{10}+5q^{8}-5q^{6}+7q^{4}-5q^{2}+4-3q^{-2}+3q^{-4}-q^{-6}+q^{-8}$
1,0	$q^{88}-q^{84}-q^{82}+q^{80}+2q^{78}-q^{76}-4q^{74}-q^{72}+4q^{70}+4q^{68}-3q^{66}-6q^{64}+6q^{60}+4q^{58}-5q^{56}-5q^{54}+q^{52}+5q^{50}-4q^{46}-q^{44}+4q^{42}+2q^{40}-3q^{38}-2q^{36}+3q^{34}+4q^{32}-2q^{30}-4q^{28}+q^{26}+5q^{24}-5q^{20}-2q^{18}+5q^{16}+5q^{14}-3q^{12}-6q^{10}-q^{8}+6q^{6}+3q^{4}-2q^{2}-3+3q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-14}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-q^{126}+2q^{124}-3q^{122}+2q^{120}-2q^{118}-2q^{116}+7q^{114}-10q^{112}+10q^{110}-10q^{108}+4q^{106}+4q^{104}-15q^{102}+20q^{100}-20q^{98}+14q^{96}-q^{94}-12q^{92}+19q^{90}-18q^{88}+15q^{86}-3q^{84}-10q^{82}+14q^{80}-11q^{78}+2q^{76}+10q^{74}-16q^{72}+18q^{70}-8q^{68}-4q^{66}+17q^{64}-26q^{62}+29q^{60}-21q^{58}+6q^{56}+11q^{54}-23q^{52}+30q^{50}-25q^{48}+13q^{46}-13q^{42}+16q^{40}-14q^{38}+3q^{36}+8q^{34}-13q^{32}+10q^{30}-2q^{28}-9q^{26}+17q^{24}-19q^{22}+15q^{20}-6q^{18}-5q^{16}+14q^{14}-16q^{12}+17q^{10}-10q^{8}+5q^{6}+q^{4}-7q^{2}+9-8q^{-2}+7q^{-4}-3q^{-6}+q^{-8}+2q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

.

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["9 12"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{2}+9t-13+9t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 35, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{8}+2z^{2}a^{6}+2a^{6}-z^{4}a^{4}-z^{2}a^{4}-a^{4}-z^{4}a^{2}-z^{2}a^{2}+z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-3z^{3}a^{9}+za^{9}+2z^{6}a^{8}-6z^{4}a^{8}+4z^{2}a^{8}-a^{8}+2z^{7}a^{7}-5z^{5}a^{7}+3z^{3}a^{7}-za^{7}+z^{8}a^{6}-5z^{4}a^{6}+7z^{2}a^{6}-2a^{6}+4z^{7}a^{5}-11z^{5}a^{5}+13z^{3}a^{5}-4za^{5}+z^{8}a^{4}-z^{4}a^{4}+3z^{2}a^{4}-a^{4}+2z^{7}a^{3}-3z^{5}a^{3}+4z^{3}a^{3}-2za^{3}+2z^{6}a^{2}-z^{4}a^{2}-2z^{2}a^{2}+2z^{5}a-3z^{3}a+z^{4}-2z^{2}+1$

Vassiliev invariants

V2 and V3:

(1, -3)

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 $4$ $-24$ $8$ ${\frac {302}{3}}$ ${\frac {58}{3}}$ $-96$ $-400$ $-32$ $-120$ ${\frac {32}{3}}$ $288$ ${\frac {1208}{3}}$ ${\frac {232}{3}}$ ${\frac {49471}{30}}$ $-{\frac {5942}{15}}$ ${\frac {49622}{45}}$ ${\frac {1793}{18}}$ ${\frac {4831}{30}}$

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

\   r    \    j   \	-7	-6	-5	-4	-3	-2	-1	0	1	2	χ
3										1	1
1									1		-1
-1								3	1		2
-3							3	2			-1
-5						3	2				1
-7					3	3					0
-9				2	3						-1
-11			1	3							2
-13		1	2								-1
-15		1									1
-17	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[9, 12]]
Out[2]=	9
In[3]:=	PD[Knot[9, 12]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 16, 6, 17], X[11, 1, 12, 18], X[17, 13, 18, 12], X[7, 14, 8, 15], X[13, 8, 14, 9], X[15, 6, 16, 7], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[9, 12]]
Out[4]=	GaussCode[-1, 9, -2, 1, -3, 8, -6, 7, -9, 2, -4, 5, -7, 6, -8, 3, -5, 4]
In[5]:=	BR[Knot[9, 12]]
Out[5]=	BR[5, {-1, -1, 2, -1, -3, 2, -3, -4, 3, -4}]
In[6]:=	alex = Alexander[Knot[9, 12]][t]
Out[6]=	2 9 2 -13 - -- + - + 9 t - 2 t 2 t t
In[7]:=	Conway[Knot[9, 12]][z]
Out[7]=	2 4 1 + z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 12], Knot[11, NonAlternating, 84]}
In[9]:=	{KnotDet[Knot[9, 12]], KnotSignature[Knot[9, 12]]}
Out[9]=	{35, -2}
In[10]:=	J=Jones[Knot[9, 12]][q]
Out[10]=	-8 2 3 5 6 6 5 4 -2 - 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[9, 12], Knot[11, NonAlternating, 15]}
In[12]:=	A2Invariant[Knot[9, 12]][q]
Out[12]=	-26 -24 -22 -18 2 -14 -10 -6 -4 2 4 -q - q + q + q + --- - q - q + q - q + -- + q 16 2 q q
In[13]:=	Kauffman[Knot[9, 12]][a, z]
Out[13]=	4 6 8 3 5 7 9 2 2 2 1 - a - 2 a - a - 2 a z - 4 a z - a z + a z - 2 z - 2 a z + 4 2 6 2 8 2 3 3 3 5 3 7 3 3 a z + 7 a z + 4 a z - 3 a z + 4 a z + 13 a z + 3 a z - 9 3 4 2 4 4 4 6 4 8 4 5 3 5 3 a z + z - a z - a z - 5 a z - 6 a z + 2 a z - 3 a z - 5 5 7 5 9 5 2 6 8 6 3 7 5 7 11 a z - 5 a z + a z + 2 a z + 2 a z + 2 a z + 4 a z + 7 7 4 8 6 8 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[9, 12]], Vassiliev[3][Knot[9, 12]]}
Out[14]=	{0, -3}
In[15]:=	Kh[Knot[9, 12]][q, t]
Out[15]=	2 3 1 1 1 2 1 3 2 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 3 3 3 3 2 3 t 3 2 ----- + ----- + ----- + ----- + ---- + ---- + - + q t + q t 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t